4.11: Vector Spaces of Random Variables
Basic Theory
Many of the concepts in this chapter have elegant interpretations if we think of real-valued random variables as vectors in a vector space. In particular, variance and higher moments are related to the concept of norm and distance, while covariance is related to inner product. These connections can help unify and illuminate some of the ideas in the chapter from a different point of view. Of course, real-valued random variables are simply measurable, real-valued functions defined on the sample space, so much of the discussion in this section is a special case of our discussion of function spaces in the chapter on Distributions, but recast in the notation of probability.
As usual, our starting point is a random experiment modeled by a probability space \( (\Omega, \mathscr{F}, \P) \). Thus, \( \Omega \) is the set of outcomes, \( \mathscr{F} \) is the \( \sigma \)-algebra of events, and \( \P \) is the probability measure on the sample space \( (\Omega, \mathscr F) \). Our basic vector space \(\mathscr V\) consists of all real-valued random variables defined on \((\Omega, \mathscr{F}, \P)\) (that is, defined for the experiment). Recall that random variables \( X_1 \) and \( X_2 \) are equivalent if \( \P(X_1 = X_2) = 1 \), in which case we write \( X_1 \equiv X_2 \). We consider two such random variables as the same vector, so that technically, our vector space consists of equivalence classes under this equivalence relation. The addition operator corresponds to the usual addition of two real-valued random variables, and the operation of scalar multiplication corresponds to the usual multiplication of a real-valued random variable by a real (non-random) number. These operations are compatible with the equivalence relation in the sense that if \( X_1 \equiv X_2 \) and \( Y_1 \equiv Y_2 \) then \( X_1 + Y_1 \equiv X_2 + Y_2 \) and \( c X_1 \equiv c X_2 \) for \( c \in \R \). In short, the vector space \( \mathscr V \) is well-defined.
Norm
Suppose that \( k \in [1, \infty) \). The \( k \) norm of \( X \in \mathscr V \) is defined by
\[ \|X\|_k = \left[\E\left(\left|X\right|^k\right)\right]^{1 / k} \]Thus, \(\|X\|_k\) is a measure of the size of \(X\) in a certain sense, and of course it's possible that \( \|X\|_k = \infty \). The following theorems establish the fundamental properties. The first is the positive property .
Suppose again that \( k \in [1, \infty) \). For \( X \in \mathscr V \),
- \(\|X\|_k \ge 0\)
- \(\|X\|_k = 0\) if and only if \(\P(X = 0) = 1\) (so that \(X \equiv 0\)).
Proof
These results follow from the basic inequality properties of expected value. First \( \left|X\right|^k \ge 0 \) with probability 1, so \( \E\left(\left|X\right|^k\right) \ge 0 \). In addition, \( \E\left(\left|X\right|^k\right) = 0 \) if and only if \( \P(X = 0) = 1 \).
The next result is the scaling property .
Suppose again that \( k \in [1, \infty) \). Then \(\|c X\|_k = \left|c\right| \, \|X\|_k\) for \( X \in \mathscr V \) and \(c \in \R\).
Proof
\[ \| c X \|_k = [\E\left(\left|c X\right|^k\right]^{1 / k} = \left[\E\left(\left|c\right|^k \left|X\right|^k\right)\right]^{1/k} = \left[\left|c\right|^k \E\left(\left|X\right|^k\right)\right]^{1/k} = \left|c\right| \left[\E\left(\left|X\right|^k\right)\right]^{1/k} = \left|c\right| \|X\|_k \]The next result is Minkowski's inequality , named for Hermann Minkowski, and also known as the triangle inequality .
Suppose again that \( k \in [1, \infty) \). Then \(\|X + Y\|_k \le \|X\|_k + \|Y\|_k\) for \( X, \, Y \in \mathscr V \).
Proof
The first quadrant \(S = \left\{(x, y) \in \R^2: x \ge 0, \; y \ge 0\right\}\) is a convex set and \(g(x, y) = \left(x ^{1/k} + y^{1/k}\right)^k\) is concave on \(S\). From Jensen's inequality, if \(U\) and \(V\) are nonnegative random variables, then \[ \E\left[(U^{1/k} + V^{1/k})^k\right] \le \left(\left[\E(U)\right]^{1/k} + \left[\E(V)\right]^{1/k}\right)^k \] Letting \(U = \left|X\right|^k\) and \(V = \left|Y\right|^k\) and simplifying gives the result. To show that \( g \) really is concave on \( S \), we can compute the second partial derivatives. Let \( h(x, y) = x^{1/k} + y^{1/k} \) so that \( g = h^k \). Then \begin{align} g_{xx} & = \frac{k-1}{k} h^{k-2} x^{1/k - 2}\left(x^{1/k} - h\right) \\ g_{yy} & = \frac{k-1}{k} h^{k-2} y^{1/k - 2}\left(y^{1/k} - h\right) \\ g_{xy} & = \frac{k-1}{k} h^{k-2} x^{1/k - 1} y^{1/k - 1} \end{align} Clearly \( h(x, y) \ge x^{1/k} \) and \( h(x, y) \ge y^{1/k} \) for \( x \ge 0 \) and \( y \ge 0 \), so \( g_{xx} \)and \( g_{yy} \), the diagonal entries of the second derivative matrix, are nonpositive on \( S \). A little algebra shows that the determinant of the second derivative matrix \( g_{xx} g_{yy} - g_{xy}^2 = 0\) on \( S \). Thus, the second derivative matrix of \( g \) is negative semi-definite.
It follows from the last three results that the set of random variables (again, modulo equivalence) with finite \(k\) norm forms a subspace of our parent vector space \(\mathscr V\), and that the \(k\) norm really is a norm on this vector space.
For \( k \in [1, \infty) \), \( \mathscr L_k \) denotes the vector space of \( X \in \mathscr V \) with \(\|X\|_k \lt \infty\), and with norm \( \| \cdot \|_k \).
In analysis, \( p \) is often used as the index rather than \( k \) as we have used here, but \( p \) seems too much like a probability, so we have broken with tradition on this point. The \( \mathscr L \) is in honor of Henri Lebesgue, who developed much of this theory. Sometimes, when we need to indicate the dependence on the underlying \( \sigma \)-algebra \( \mathscr{F} \), we write \( \mathscr L_k(\mathscr{F}) \). Our next result is Lyapunov's inequality , named for Aleksandr Lyapunov. This inequality shows that the \(k\)-norm of a random variable is increasing in \(k\).
Suppose that \( j, \, k \in [1, \infty) \) with \(j \le k\). Then \(\|X\|_j \le \|X\|_k\) for \(X \in \mathscr V\).
Proof
Note that \(S = \{x \in \R: x \ge 0\}\) is convex and \(g(x) = x^{k/j}\) is convex on \(S\). From Jensen's inequality, if \(U\) is a nonnegative random variable then \(\left[\E(U)\right]^{k/j} \le \E\left(U^{k/j}\right)\). Letting \(U = \left|X\right|^j\) and simplifying gives the result.
Lyapunov's inequality shows that if \(1 \le j \le k\) and \( \|X\|_k \lt \infty \) then \( \|X\|_j \lt \infty \). Thus, \(\mathscr L_k\) is a subspace of \(\mathscr L_j\).
Metric
The \(k\) norm, like any norm on a vector space, can be used to define a metric, or distance function; we simply compute the norm of the difference between two vectors.
For \( k \in [1, \infty) \), the \(k\) distance (or \(k\) metric ) between \(X, \, Y \in \mathscr V\) is defined by \[ d_k(X, Y) = \|X - Y\|_k = \left[\E\left(\left|X - Y\right|^k\right)\right]^{1/k} \]
The following properties are analogous to the properties in norm properties (and thus very little additional work is required for the proofs). These properties show that the \(k\) metric really is a metric on \( \mathscr L_k \) (as always, modulo equivalence). The first is the positive property .
Suppose again that \( k \in [1, \infty) \) \(X, \; Y \in \mathscr V\). Then
- \(d_k(X, Y) \ge 0\)
- \(d_k(X, Y) = 0\) if and only if \(\P(X = Y) = 1\) (so that \(X \equiv Y\) and \(Y\)).
Proof
These results follow directly from the positive property .
Next is the obvious symmetry property :
\( d_k(X, Y) = d_k(Y, X) \) for \( X, \; Y \in \mathscr V \).
Next is the distance version of the triangle inequality .
\(d_k(X, Z) \le d_k(X, Y) + d_k(Y, Z)\) for \(X, \; Y, \; Z \in \mathscr V\)
Proof
From Minkowski's inequality , \[ d_k(X, Z) = \|X - Z\|_k = \|(X - Y) + (Y - Z) \|_k \le \|X - Y\|_k + \|Y - Z\|_k = d_k(X, Y) + d_k(Y, Z) \]
The last three properties mean that \( d_k \) is indeed a metric on \( \mathscr L_k \) for \( k \ge 1 \). In particular, note that the standard deviation is simply the 2-distance from \(X\) to its mean \( \mu = \E(X) \): \[ \sd(X) = d_2(X, \mu) = \|X - \mu\|_2 = \sqrt{\E\left[(X - \mu)^2\right]} \] and the variance is the square of this. More generally, the \(k\)th moment of \(X\) about \(a\) is simply the \(k\)th power of the \(k\)-distance from \(X\) to \(a\). The 2-distance is especially important for reasons that will become clear below, in the discussion of inner product . This distance is also called the root mean square distance .
Center and Spread Revisited
Measures of center and measures of spread are best thought of together, in the context of a measure of distance . For a real-valued random variable \(X\), we first try to find the constants \(t \in \R\) that are closest to \(X\), as measured by the given distance; any such \(t\) is a measure of center relative to the distance. The minimum distance itself is the corresponding measure of spread .
Let us apply this procedure to the 2-distance.
For \( X \in \mathscr L_2 \), define the root mean square error function by \[ d_2(X, t) = \|X - t\|_2 = \sqrt{\E\left[(X - t)^2\right]}, \quad t \in \R \]
For \( X \in \mathscr L_2 \), \(d_2(X, t)\) is minimized when \(t = \E(X)\) and the minimum value is \(\sd(X)\).
Proof
Note that the minimum value of \(d_2(X, t)\) occurs at the same points as the minimum value of \(d_2^2(X, t) = \E\left[(X - t)^2\right]\) (this is the mean square error function). Expanding and taking expected values term by term gives \[ \E\left[(X - t)^2\right] = \E\left(X^2\right) - 2 t \E(X) + t^2 \] This is a quadratic function of \( t \) and hence the graph is a parabola opening upward. The minimum occurs at \( t = \E(X) \), and the minimum value is \( \var(X) \). Hence the minimum value of \( t \mapsto d_2(X, t) \) also occurs at \( t = \E(X) \) and the minimum value is \( \sd(X) \).
We have seen this computation several times before. The best constant predictor of \( X \) is \( \E(X) \), with mean square error \( \var(X) \). The physical interpretation of this result is that the moment of inertia of the mass distribution of \(X\) about \(t\) is minimized when \(t = \mu\), the center of mass. Next, let us apply our procedure to the 1-distance.
For \( X \in \mathscr L_1 \), define the mean absolute error function by \[ d_1(X, t) = \|X - t\|_1 = \E\left[\left|X - t\right|\right], \quad t \in \R \]
We will show that \(d_1(X, t)\) is minimized when \(t\) is any median of \(X\). (Recall that the set of medians of \( X \) forms a closed, bounded interval.) We start with a discrete case, because it's easier and has special interest.
Suppose that \(X \in \mathscr L_1\) has a discrete distribution with values in a finite set \(S \subseteq \R\). Then \(d_1(X, t)\) is minimized when \(t\) is any median of \(X\).
Proof
Note first that \(\E\left(\left|X - t\right|\right) = \E(t - X, \, X \le t) + \E(X - t, \, X \gt t)\). Hence \(\E\left(\left|X - t\right|\right) = a_t \, t + b_t\), where \(a_t = 2 \, \P(X \le t) - 1\) and where \(b_t = \E(X) - 2 \, \E(X, \, X \le t)\). Note that \(\E\left(\left|X - t\right|\right)\) is a continuous, piecewise linear function of \(t\), with corners at the values in \(S\). That is, the function is a linear spline . Let \(m\) be the smallest median of \(X\). If \(t \lt m\) and \(t \notin S\), then the slope of the linear piece at \(t\) is negative. Let \(M\) be the largest median of \(X\). If \(t \gt M\) and \(t \notin S\), then the slope of the linear piece at \(t\) is positive. If \(t \in (m, M)\) then the slope of the linear piece at \(t\) is 0. Thus \(\E\left(\left|X - t\right|\right)\) is minimized for every \(t\) in the median interval \([m, M]\).
The last result shows that mean absolute error has a couple of basic deficiencies as a measure of error:
- The function may not be smooth (differentiable).
- The function may not have a unique minimizing value of \(t\).
Indeed, when \(X\) does not have a unique median, there is no compelling reason to choose one value in the median interval, as the measure of center, over any other value in the interval.
Suppose now that \(X \in \mathscr L_1 \) has a general distribution on \(\R\). Then \(d_1(X, t)\) is minimized when \(t\) is any median of \(X\).
Proof
Let \( s, \, t \in \R \). Suppose first that \(s \lt t\). Computing the expected value over the events \(X \le s\), \(s \lt X \le t\), and \(X \ge t\), and simplifying gives \[ \E\left(\left|X - t\right|\right) = \E\left(\left|X - s\right|\right) + (t - s) \, \left[2 \, \P(X \le s) - 1\right] + 2 \, \E(t - X, \, s \lt X \le t) \] Suppose next that \(t \lt s\). Using similar methods gives \[ \E\left(\left|X - t\right|\right) = \E\left(\left|X - s\right|\right) + (t - s) \, \left[2 \, \P(X \lt s) - 1\right] + 2 \, \E(X - t, \, t \le X \lt s) \] Note that the last terms on the right in these equations are nonnegative. If we take \(s\) to be a median of \(X\), then the middle terms on the right in the equations are also nonnegative. Hence if \(s\) is a median of \(X\) and \(t\) is any other number then \(\E\left(\left|X - t\right|\right) \ge \E\left(\left|X - s\right|\right)\).
Convergence
Whenever we have a measure of distance, we automatically have a criterion for convergence.
Suppose that \( X_n \in \mathscr L_k \) for \( n \in \N_+ \) and that \( X \in \mathscr L_k \), where \( k \in [1, \infty) \). Then \(X_n \to X\) as \(n \to \infty\) in \(k\)th mean if \( X_n \to X \) as \( n \to \infty \) in the vector space \( \mathscr L_k \). That is, \[ d_k(X_n, X) = \|X_n - X\|_k \to 0 \text{ as } n \to \infty \] or equivalently \( \E\left(\left|X_n - X\right|^k\right) \to 0\) as \(n \to \infty \).
When \(k = 1\), we simply say that \(X_n \to X\) as \(n \to \infty\) in mean ; when \(k = 2\), we say that \(X_n \to X\) as \(n \to \infty\) in mean square . These are the most important special cases.
Suppose that \(1 \le j \le k\). If \(X_n \to X\) as \(n \to \infty\) in \(k\)th mean then \(X_n \to X\) as \(n \to \infty\) in \(j\)th mean.
Proof
This follows from Lyanpuov's inequality . Note that \( 0 \le d_j(X_n, X) \le d_k(X_n, X) \to 0 \) as \( n \to \infty \).
Convergence in \( k \)th mean implies that the \( k \) norms converge.
Suppose that \( X_n \in \mathscr L_k \) for \( n \in \N_+ \) and that \( X \in \mathscr L_k \), where \( k \in [1, \infty) \). If \( X_n \to X \) as \( n \to \infty \) in \( k \)th mean then \( \|X_n\|_k \to \|X\|_k \) as \( n \to \infty \). Equivalently, if \( \E(|X_n - X|^k) \to 0 \) as \( n \to \infty \) then \( \E(|X_n|^k) \to \E(|X|^k) \) as \( n \to \infty \).
Proof
This is a simple consequence of the reverse triangle inequality, which holds in any normed vector space. The general result is that if a sequence of vectors in a normed vector space converge then the norms converge. In our notation here, \[ \left|\|X_n\|_k - \|X\|_k\right| \le \|X_n - X\|_k \] so if the right side converges to 0 as \( n \to \infty \), then so does the left side.
The converse is not true; a counterexample is given below. Our next result shows that convergence in mean is stronger than convergence in probability.
Suppose that \( X_n \in \mathscr L_1 \) for \( n \in \N_+ \) and that \( X \in \mathscr L_1 \). If \(X_n \to X\) as \(n \to \infty\) in mean, then \(X_n \to X\) as \(n \to \infty\) in probability.
Proof
This follows from Markov's inequality. For \( \epsilon \gt 0 \), \(0 \le \P\left(\left|X_n - X\right| \gt \epsilon\right) \le \E\left(\left|X_n - X\right|\right) \big/ \epsilon \to 0 \) as \( n \to \infty \).
The converse is not true. That is, convergence with probability 1 does not imply convergence in \(k\)th mean; a counterexample is given below. Also convergence in \(k\)th mean does not imply convergence with probability 1; a counterexample to this is given below. In summary, the implications in the various modes of convergence are shown below; no other implications hold in general.
- Convergence with probability 1 implies convergence in probability.
- Convergence in \(k\)th mean implies convergence in \(j\)th mean if \(j \le k\).
- Convergence in \(k\)th mean implies convergence in probability.
- Convergence in probability implies convergence in distribution.
However, the next section on uniformly integrable variables gives a condition under which convergence in probability implies convergence in mean.
Inner Product
The vector space \( \mathscr L_2 \) of real-valued random variables on \( (\Omega, \mathscr{F}, \P) \) (modulo equivalence of course) with finite second moment is special, because it's the only one in which the norm corresponds to an inner product.
The inner product of \( X, \, Y \in \mathscr L_2 \) is defined by \[ \langle X, Y \rangle = \E(X Y) \]
The following results are analogous to the basic properties of covariance, and show that this definition really does give an inner product on the vector space
For \( X, \, Y, \, Z \in \mathscr L_2 \) and \( a \in \R \),
- \(\langle X, Y \rangle = \langle Y, X \rangle\), the symmetric property .
- \(\langle X, X \rangle \ge 0\) and \(\langle X, X \rangle = 0\) if and only if \(\P(X = 0) = 1\) (so that \(X \equiv 0\)), the positive property .
- \(\langle a X, Y \rangle = a \langle X, Y \rangle\), the scaling property .
- \(\langle X + Y, Z \rangle = \langle X, Z \rangle + \langle Y, Z \rangle\), the additive property .
Proof
- This property is trivial from the definition.
- Note that \( \E(X^2) \ge 0 \) and \( \E(X^2) = 0 \) if and only if \( \P(X = 0) = 1 \).
- This follows from the scaling property of expected value: \( \E(a X Y) = a \E(X Y) \)
- This follows from the additive property of expected value: \( \E[(X + Y) Z] = \E(X Z) + \E(Y Z) \).
From parts (a), (c), and (d) it follows that inner product is bi-linear , that is, linear in each variable with the other fixed. Of course bi-linearity holds for any inner product on a vector space. Covariance and correlation can easily be expressed in terms of this inner product. The covariance of two random variables is the inner product of the corresponding centered variables. The correlation is the inner product of the corresponding standard scores.
For \( X, \, Y \in \mathscr L_2 \),
- \(\cov(X, Y) = \langle X - \E(X), Y - \E(Y) \rangle\)
- \(\cor(X, Y) = \left \langle [X - \E(X)] \big/ \sd(X), [Y - \E(Y)] / \sd(Y) \right \rangle\)
Proof
- This is simply a restatement of the definition of covariance.
- This is a restatement of the fact that the correlation of two variables is the covariance of their corresponding standard scores.
Thus, real-valued random variables \( X \) and \( Y \) are uncorrelated if and only if the centered variables \( X - \E(X) \) and \( Y - \E(Y) \) are perpendicular or orthogonal as elements of \( \mathscr L_2 \).
For \( X \in \mathscr L_2 \), \(\langle X, X \rangle = \|X\|_2^2 = \E\left(X^2\right)\).
Thus, the norm associated with the inner product is the 2-norm studied above, and corresponds to the root mean square operation on a random variable. This fact is a fundamental reason why the 2-norm plays such a special, honored role; of all the \(k\)-norms, only the 2-norm corresponds to an inner product. In turn, this is one of the reasons that root mean square difference is of fundamental importance in probability and statistics. Technically, the vector space \( \mathscr L_2 \) is a Hilbert space , named for David Hilbert.
The next result is Hölder's inequality , named for Otto Hölder.
Suppose that \(j, \, k \in [1, \infty)\) and \(\frac{1}{j} + \frac{1}{k} = 1\). For \( X \in \mathscr L_j \) and \( Y \in \mathscr L_k \), \[\langle \left|X\right|, \left|Y\right| \rangle \le \|X\|_j \|Y\|_k \]
Proof
Note that \(S = \left\{(x, y) \in \R^2: x \ge 0, \; y \ge 0\right\}\) is a convex set and \(g(x, y) = x^{1/j} y^{1/k}\) is concave on \(S\). From Jensen's inequality, if \(U\) and \(V\) are nonnegative random variables then \(\E\left(U^{1/j} V^{1/k}\right) \le \left[\E(U)\right]^{1/j} \left[\E(V)\right]^{1/k}\). Substituting \(U = \left|X\right|^j\) and \(V = \left|Y\right|^k\) gives the result.
To show that \( g \) really is concave on \( S \), we compute the second derivative matrix:
\[ \left[ \begin{matrix} (1 / j)(1 / j - 1) x^{1 / j - 2} y^{1 / k} & (1 / j)(1 / k) x^{1 / j - 1} y^{1 / k - 1} \\ (1 / j)(1 / k) x^{1 / j - 1} y^{1 / k - 1} & (1 / k)(1 / k - 1) x^{1 / j} y^{1 / k - 2} \end{matrix} \right] \]Since \( 1 / j \lt 1 \) and \( 1 / k \lt 1 \), the diagonal entries are negative on \( S \). The determinant simplifies to
\[ (1 / j)(1 / k) x^{2 / j - 2} y^{2 / k - 2} [1 - (1 / j + 1 / k)] = 0 \]In the context of the last theorem, \(j\) and \(k\) are called conjugate exponents . If we let \(j = k = 2\) in Hölder's inequality, then we get the Cauchy-Schwarz inequality , named for Augustin Cauchy and Karl Schwarz: For \( X, \, Y \in \mathscr L_2 \), \[ \E\left(\left|X\right| \left|Y\right|\right) \le \sqrt{\E\left(X^2\right)} \sqrt{\E\left(Y^2\right)} \] In turn, the Cauchy-Schwarz inequality is equivalent to the basic inequalities for covariance and correlations: For \( X, \, Y \in \mathscr L_2 \), \[ \left| \cov(X, Y) \right| \le \sd(X) \sd(Y), \quad \left|\cor(X, Y)\right| \le 1 \]
If \(j, \, k \in [1, \infty)\) are conjugate exponents then
- \(k = \frac{j}{j - 1}\).
- \(k \downarrow 1\) as \(j \uparrow \infty\).
The following result is an equivalent to the identity \( \var(X + Y) + \var(X - Y) = 2\left[\var(X) + \var(Y)\right] \) that we studied in the section on covariance and correlation. In the context of vector spaces, the result is known as the parallelogram rule :
If \(X, \, Y \in \mathscr L_2\) then \[ \|X + Y\|_2^2 + \|X - Y\|_2^2 = 2 \|X\|_2^2 + 2 \|Y\|_2^2\]
Proof
This result follows from the bi-linearity of inner product: \begin{align} \|X + Y\|_2^2 + \|X - Y\|_2^2 & = \langle X + Y, X + Y \rangle + \langle X - Y, X - Y\rangle \\ & = \left(\langle X, X \rangle + 2 \langle X, Y \rangle + \langle Y, Y \rangle\right) + \left(\langle X, X \rangle - 2 \langle X, Y \rangle + \langle Y, Y \rangle\right) = 2 \|X\|^2 + 2 \|Y\|^2 \end{align}
The following result is equivalent to the statement that the variance of the sum of uncorrelated variables is the sum of the variances, which again we proved in the section on covariance and correlation. In the context of vector spaces, the result is the famous Pythagorean theorem , named for Pythagoras of course.
If \((X_1, X_2, \ldots, X_n)\) is a sequence of random variables in \(\mathscr L_2\) with \(\langle X_i, X_j \rangle = 0\) for \(i \ne j\) then \[ \left \| \sum_{i=1}^n X_i \right \|_2^2 = \sum_{i=1}^n \|X_i\|_2^2 \]
Proof
Again, this follows from the bi-linearity of inner product: \[ \left \| \sum_{i=1}^n X_i \right \|_2^2 = \left\langle \sum_{i=1}^n X_i, \sum_{j=1}^n X_j\right\rangle = \sum_{i=1}^n \sum_{j=1}^n \langle X_i, X_j \rangle \] The terms with \( i \ne j \) are 0 by the orthogonality assumption, so \[ \left \| \sum_{i=1}^n X_i \right \|_2^2 = \sum_{i=1}^n \langle X_i, X_i \rangle = \sum_{i=1}^n \|X_i\|_2^2 \]
Projections
The best linear predictor studied in the section on covariance and correlation and conditional expected values have nice interpretation in terms of projections onto subspaces of \( \mathscr L_2 \). First let's review the concepts.  To show that \( \mathscr U \subseteq \mathscr L_2 \) is a subspace, we just need to show the closure properties (the other axioms of a vector space are inherited).

  If you look at the proofs of these results, you will see that they are essentially the same as the ones used for the best predictors of \( X \) mentioned at the beginning of this subsection. Moreover, the proofs use only vector space concepts—the fact that our vectors are random variables on a probability space plays no special role.
The projection of \( X \) onto \( \mathscr U \) (if it exists) is unique.
Proof
Suppose that \( V_1 \) and \( V_2 \) satisfy the definition. then \[ \left\|V_1 - V_2\right\|_2^2 = \langle V_1 - V_2, V_1 - V_2 \rangle = \langle V_1 - X + X - V_2, V_1 - V_2 \rangle = \langle V_1 - X, V_1 - V_2 \rangle + \langle X - V_2, V_1 - V_2 \rangle = 0 \] Hence \( V_1 \equiv V_2 \).  Then
- \( \left\|X - V\right\|_2^2 \le \left\|X - U\right\|_2^2\) for all \( U \in \mathscr U \).
- Equality holds in (a) if and only if \( U \equiv V \)
Proof
- If \( U \in \mathscr U \) then \[ \left\| X - U \right\|_2^2 = \left\| X - V + V - U \right\|_2^2 = \left\| X - V \right\|_2^2 + 2 \langle X - V, V - U \rangle + \left\| V - U \right\|_2^2\] But the middle terms is 0 so \[ \left\| X - U \right\|_2^2 = \left\| X - V \right\|_2^2 + \left\| V - U \right\|_2^2 \ge \left\| X - V \right\|_2^2\]
- Equality holds if and only if \( \left\| V - U \right\|_2^2 = 0\), if and only if \( V \equiv U \).
Now let's return to our study of best predictors of a random variable.
If \( X \in \mathscr L_2 \) then the set \( \mathscr W_X = \{a + b X: a \in \R, \; b \in \R\} \) is a subspace of \(\mathscr L_2\). In fact, it is the subspace generated by \(X\) and 1.
Proof
Note that \( \mathscr W_X \) is the set of all linear combinations of the vectors \( 1 \) and \( X \). If \( U, \, V \in \mathscr W_X \) then \( U + V \in \mathscr W_X \). If \( U \in \mathscr W_X \) and \( c \in \R \) then \( c U \in \mathscr W_X \).
Recall that for \( X, \, Y \in \mathscr L_2 \), the best linear predictor of \( Y \) based on \( X \) is \[ L(Y \mid X) = \E(Y) + \frac{\cov(X, Y)}{\var(X)} \left[X - \E(X)\right] \] Here is the meaning of the predictor in the context of our vector spaces.
If \( X, \, Y \in \mathscr L_2 \) then \( L(Y \mid X) \) is the projection of \(Y\) onto \(\mathscr W_X\).
Proof
Note first that \(L(Y \mid X) \in \mathscr W_X \). Thus, we just need to show that \( Y - L(Y \mid X) \) is perpendicular to \( \mathscr W_X \). For this, it suffices to show
- \(\left\langle Y - L(Y \mid X), X \right\rangle = 0\)
- \(\left\langle Y - L(Y \mid X), 1 \right\rangle = 0\)
We have already done this in the earlier sections, but for completeness, we do it again. Note that \( \E\left(X \left[X - \E(X)\right]\right) = \var(X) \). Hence \( \E\left[X L(Y \mid X)\right] = \E(X) \E(Y) + \cov(X, Y) = \E(X Y) \). This gives (a). By linearity, \( \E\left[L(Y \mid X)\right] = \E(Y) \) so (b) holds as well.
The previous result is actually just the random variable version of the standard formula for the projection of a vector onto a space spanned by two other vectors. Note that \( 1 \) is a unit vector and that \( X_0 = X - \E(X) = X - \langle X, 1 \rangle 1 \) is perpendicular to \( 1 \). Thus, \( L(Y \mid X) \) is just the sum of the projections of \( Y \) onto \( 1 \) and \( X_0 \): \[ L(Y \mid X) = \langle Y, 1 \rangle 1 + \frac{\langle Y, X_0 \rangle}{\langle X_0, X_0\rangle} X_0 \]
Suppose now that \( \mathscr{G} \) is a sub \( \sigma \)-algebra of \( \mathscr{F} \). Of course if \( X: \Omega \to \R \) is \( \mathscr{G} \)-measurable then \( X \) is \( \mathscr{F} \)-measurables, so \( \mathscr L_2(\mathscr{G}) \) is a subspace of \( \mathscr L_2(\mathscr{F}) \).
If \( X \in \mathscr L_2(\mathscr{F}) \) then \( \E(X \mid \mathscr{G}) \) is the projection of \( X \) onto \( \mathscr L_2(\mathscr{G}) \).
Proof
This is essentially the definition of \( \E(X \mid \mathscr{G}) \) as the only (up to equivalence) random variable in \( \mathscr L_2(\mathscr{G}) \) with \( \E\left[\E(X \mid \mathscr{G}) U\right] = \E(X U) \) for every \( U \in \mathscr L_2(\mathscr{G}) \).
But remember that \( \E(X \mid \mathscr{G}) \) is defined more generally for \( X \in \mathscr L_1(\mathscr{F}) \). Our final result in this discussion concerns convergence.
Suppose that \( k \in [1, \infty) \) and that \( \mathscr{G} \) is a sub \( \sigma \)-algebra of \( \mathscr{F} \).
- If \( X \in \mathscr L_k(\mathscr{F}) \) then \( \E(X \mid \mathscr{G}) \in \mathscr L_k(\mathscr{G}) \)
- If \( X_n \in \mathscr L_k(\mathscr{F}) \) for \( n \in \N_+ \), \( X \in \mathscr L_k(\mathscr{F}) \), and \( X_n \to X \) as \( n \to \infty \) in \( \mathscr L_k(\mathscr{F}) \) then \( \E(X_n \mid \mathscr{G}) \to \E(X \mid \mathscr{G}) \) as \( n \to \infty \) in \( \mathscr L_k(\mathscr{G}) \)
Proof
- Note that \( |\E(X \mid \mathscr{G})| \le \E(|X| \mid \mathscr{G}) \). Since \( t \mapsto t^k \) is increasing and convex on \( [0, \infty) \) we have \[ |\E(X \mid \mathscr{G})|^k \le [\E(|X| \mid \mathscr{G})]^k \le \E\left(|X|^k \mid \mathscr{G}\right) \] The last step uses Jensen's inequality. Taking expected values gives \[ \E[|\E(X \mid \mathscr{G})|^k] \le \E(|X|^k) \lt \infty \]
- Using the same ideas, \[ \E\left[\left|\E(X_n \mid \mathscr{G}) - \E(X \mid \mathscr{G})\right|^k\right] = \E\left[\left|\E(X_n - X \mid \mathscr{G})\right|^k\right] \le E[|X_n - X|^k] \] By assumption, the right side converges to 0 as \( n \to \infty \) and hence so does the left side.
Examples and Applications
App Exercises
In the error function app, select the root mean square error function. Click on the \( x \)-axis to generate an empirical distribution, and note the shape and location of the graph of the error function.
In the error function app, select the mean absolute error function. Click on the \( x \)-axis to generate an empirical distribution, and note the shape and location of the graph of the error function.
Computational Exercises
Suppose that \(X\) is uniformly distributed on the interval \([0, 1]\).
- Find \(\|X\|_k\) for \( k \in [1, \infty) \).
- Graph \(\|X\|_k\) as a function of \(k \in [1, \infty)\).
- Find \(\lim_{k \to \infty} \|X\|_k\).
Answer
- \(\frac{1}{(k + 1)^{1/k}}\)
- 1
Suppose that \(X\) has probability density function \(f(x) = \frac{a}{x^{a+1}}\) for \(1 \le x \lt \infty\), where \(a \gt 0\) is a parameter. Thus, \(X\) has the Pareto distribution with shape parameter \(a\).
- Find \(\|X\|_k\) for \( k \in [1, \infty) \).
- Graph \(\|X\|_k\) as a function of \(k \in (1, a)\).
- Find \(\lim_{k \uparrow a} \|X\|_k\).
Answer
- \(\left(\frac{a}{a -k}\right)^{1/k}\) if \(k \lt a\), \(\infty\) if \(k \ge a\)
- \(\infty\)
Suppose that \((X, Y)\) has probability density function \(f(x, y) = x + y\) for \(0 \le x \le 1\), \(0 \le y \le 1\). Verify Minkowski's inequality.
Answer
- \(\|X + Y\|_k = \left(\frac{2^{k+2} - 2}{(k + 2)(k + 3)}\right)^{1/k}\)
- \(\|X\|_k + \|Y\|_k = 2 \left(\frac{1}{k + 2} + \frac{1}{2(k + 1)}\right)^{1/k}\)
Let \(X\) be an indicator random variable with \(\P(X = 1) = p\), where \(0 \le p \le 1\). Graph \(\E\left(\left|X - t\right|\right)\) as a function of \(t \in \R\) in each of the cases below. In each case, find the minimum value of the function and the values of \(t\) where the minimum occurs.
- \(p \lt \frac{1}{2}\)
- \(p = \frac{1}{2}\)
- \(p \gt \frac{1}{2}\)
Answer
- The minimum is \(p\) and occurs at \(t = 0\).
- The minimum is \(\frac{1}{2}\) and occurs for \(t \in [0, 1]\)
- The minimum is \(1 - p\) and occurs at \(t = 1\)
Suppose that \(X\) is uniformly distributed on the interval \([0, 1]\). Find \(d_1(X, t) = \E\left(\left|X - t\right|\right)\) as a function of \(t\) and sketch the graph. Find the minimum value of the function and the value of \(t\) where the minimum occurs.
Suppose that \(X\) is uniformly distributed on the set \([0, 1] \cup [2, 3]\). Find \(d_1(X, t) = \E\left(\left|X - t\right|\right)\) as a function of \(t\) and sketch the graph. Find the minimum value of the function and the values of \(t\) where the minimum occurs.
Suppose that \((X, Y)\) has probability density function \(f(x, y) = x + y\) for \(0 \le x \le 1\), \(0 \le y \le 1\). Verify Hölder's inequality in the following cases:
- \(j = k = 2\)
- \(j = 3\), \(k = \frac{3}{2}\)
Answer
- \(\|X\|_2 \|Y\|_2 = \frac{5}{12}\)
- \(\|X\|_3 + \|Y\|_{3/2} \approx 0.4248\)
Counterexamples
The following exercise shows that convergence with probability 1 does not imply convergence in mean.
Suppose that \((X_1, X_2, \ldots)\) is a sequence of independent random variables with \[ \P\left(X = n^3\right) = \frac{1}{n^2}, \; \P(X_n = 0) = 1 - \frac{1}{n^2}; \quad n \in \N_+ \]
- \(X_n \to 0\) as \(n \to \infty\) with probability 1.
- \(X_n \to 0\) as \(n \to \infty\) in probability.
- \(\E(X_n) \to \infty\) as \(n \to \infty\).
Proof
- This follows from the basic characterization of convergence with probability 1: \( \sum_{n=1}^\infty \P(X_n \gt \epsilon) = \sum_{n=1}^\infty 1 / n^2 \lt \infty \) for \( 0 \lt \epsilon \lt 1 \).
- This follows since convergence with probability 1 implies convergence in probability.
- Note that \( \E(X_n) = n^3 / n^2 = n \) for \( n \in \N_+ \).
The following exercise shows that convergence in mean does not imply convergence with probability 1.
Suppose that \((X_1, X_2, \ldots)\) is a sequence of independent indicator random variables with \[ \P(X_n = 1) = \frac{1}{n}, \; \P(X_n = 0) = 1 - \frac{1}{n}; \quad n \in \N_+ \]
- \(\P(X_n = 0 \text{ for infinitely many } n) = 1\).
- \(\P(X_n = 1 \text{ for infinitely many } n) = 1\).
- \(\P(X_n \text{ does not converge as } n \to \infty) = 1\).
- \(X_n \to 0\) as \(n \to \infty\) in \(k\)th mean for every \(k \ge 1\).
Proof
- This follows from the second Borel-Cantelli lemma since \( \sum_{n=1}^\infty \P(X_n = 1) = \sum_{n=1}^\infty 1 / n = \infty \)
- This also follows from the second Borel-Cantelli lemma since \( \sum_{n=1}^\infty \P(X_n = 0) = \sum_{n=1}^\infty (1 - 1 / n) = \infty \).
- This follows from parts (a) and (b).
- Note that \( \E(X_n) = 1 / n \to 0 \) as \( n \to \infty \).
The following exercise show that convergence of the \( k \)th means does not imply convergence in \( k \)th mean.
Suppose that \( U \) has the Bernoulli distribution with parmaeter \( \frac{1}{2} \), so that \( \P(U = 1) = \P(U = 0) = \frac{1}{2} \). Let \( X_n = U \) for \( n \in \N_+ \) and let \( X = 1 - U \). Let \( k \in [1, \infty) \). Then
- \( \E(X_n^k) = \E(X^k) = \frac{1}{2} \) for \( n \in \N_+ \), so \( \E(X_n^k) \to \E(X^k) \) as \( n \to \infty \)
- \( \E(|X_n - X|^k) = 1 \) for \( n \in \N \) so \( X_n \) does not converge to \( X \) as \( n \to \infty \) in \( \mathscr L_k \).
Proof
- Note that \( X_n^k = U^k = U \) for \( n \in \N_+ \), since \( U \) just takes values 0 and 1. Also, \( U \) and \( 1 - U \) have the same distribution so \( \E(U) = \E(1 - U) = \frac{1}{2} \).
- Note that \( X_n - X = U - (1 - U) = 2 U - 1 \) for \( n \in \N_+ \). Again, \( U \) just takes values 0 and 1, so \( |2 U - 1| = 1 \).


3: Spatial Gradients in Biodiversity
Generally speaking, warm tropical ecosystems are richer in species than cold temperate ecosystems at high latitudes (see Gaston and Williams, 1996, for general discussion). A similar pattern is seen for higher taxonomic groups (genera, families). Various hypotheses (e.g., environmental patchiness, solar energy, productivity; see Blackburn and Gaston, 1996) have been raised to explain these patterns. For example, it is assumed that warm, moist, tropical environments, with long day-lengths provide organisms with more resources for growth and reproduction than harsh environments with low energy resources (Hunter, 2002). When environmental conditions favor the growth and reproduction of primary producers (e.g., aquatic algae, corals, terrestrial flora) then these may support large numbers of secondary consumers, such as small herbivores, which also support a more numerous and diverse fauna of predators. In contrast, the development of primary producers in colder temperate ecosystems is constrained by seasonal changes in sunlight and temperature. Consequently, these ecosystems may support a less diverse biota of secondary consumers and predators.
Recently, (Allen et al. 2002) developed a model for the effect of ambient temperature on metabolism, and hence generation time and speciation rates, and used this model to explain the latitudinal gradient in biodiversity. However, these authors also noted that the principles that underlie these spatial pattern of biodiversity are still not well understood.
Species and ecosystem diversity is also known to vary with altitude Walter (1985) and Gaston and Williams (1996: 214-215). Mountainous environments, also called orobiomes, are subdivided vertically into altitudinal belts, such as montane, alpine and nival, that have quite different ecosystems. Climatic conditions at higher elevations (e.g., low temperatures, high aridity) can create environments where relatively few species can survive. Similarly, in oceans and freshwaters there are usually fewer species as one moves to increasing depths below the surface. However, in the oceans there may be a rise in species richness close to the seabed, which is associated with an increase in ecosystem heterogeneity.
By mapping spatial gradients in biodiversity we can also identify areas of special conservation interest. Conservation biologists are interested in areas that have a high proportion of endemic species, i.e., species whose distributions are naturally restricted to a limited area. It is obviously important to conserve these areas because much of their flora and fauna, and therefore the ecosystems so-formed, are found nowhere else. Areas of high endemism are also often associated with high species richness (see Gaston and Spicer, 1998 for references).
Some conservation biologists have focused their attention on areas that have high levels of endemism (and hence diversity) that are also experiencing a high rate of loss of ecosystems; these regions are biodiversity hotspots. Because biodiversity hotspots are characterized by localized concentrations of biodiversity under threat, they represent priorities for conservation action (Sechrest et al., 2002). A terrestrial biodiversity hotspot is defined quantitatively as an area that has at least 0.5%, or 1,500 of the world's ca. 300,000 species of green plants (Viridiplantae), and that has lost at least 70% of its primary vegetation (Myers et al., 2000; Conservation International, 2002). Marine biodiversity hotspots are quantitatively defined based on measurements of relative endemism of multiple taxa (species of corals, snails, lobsters, fishes) within a region and the relative level of threat to that region (Roberts et al., 2002). According to this approach, the Philippine archipelago and the islands of Bioko, Sao Tome, Principe and Annobon in the eastern Atlantic Gulf of Guinea are ranked as two of the most threatened marine biodiversity hotspot regions.
Conservation biologists may also be interested in biodiversity coldspots; these are areas that have relatively low biological diversity but also include threatened ecosystems (Kareiva and Marvier, 2003). Although a biodiversity coldspot is low in species richness, it can also be important to conserve, as it may be the only location where a rare species is found. Extreme physical environments (low or high temperatures or pressures, or unusual chemical composition) inhabited by just one or two specially adapted species are coldspots that warrant conservation because they represent unique environments that are biologically and physically interesting. For further discussion on spatial gradients in biodiversity and associated conservation practices see the related modules on "Where is the world's biodiversity?" and "Conservation Planning at a Regional Scale."
Glossary
- Biodiversity hotspots
- in general terms these are areas that have high levels of endemism (and hence diversity) but which are also experiencing a high rate of loss of habitat. This concept was originally developed for terrestrial ecosystems. A terrestrial biodiversity hotspot is an area that has at least 0.5%, or 1,500 of the worlds ca. 300,000 species of green plants (Viridiplantae), and that has lost at least 70% of its primary vegetation (Myers et al., 2000). Marine biodiversity hotspots have been defined for coral reefs, based on measurements of relative endemism of multiple taxa (species of corals, snails, lobsters, fishes) within a region and the relative level of threat to that region (Roberts et al., 2002)
- Orobiome
- a mountainous environment or landscape with its constituent ecosystems
- Species richness
- the number of different species in a particular area.
- ecosystem
- a community plus the physical environment that it occupies at a given time.
- Area of endemism
- an areas which has a high proportion of endemic species (i.e., species with distributions that are naturally restricted to that region)
- Endemic species
- those species whose distributions are naturally restricted to a defined region
- Terrestrial Biodiversity hotspots
- Marine Biodiversity hotspots
- Biodiversity coldspots
- areas that have relatively low biological diversity but are also experiencing a high rate of habitat loss


7.5: Práctica Los verbos reflexivos. Add Text only
-
- Last updated
- Save as PDF
- Romano Sánchez Domínguez
- Imperial Valley College
Reflexive verbs
In the examples below, pay special attention to the reflexive pronouns and the conjugation endings noted in blue.
In Spanish, to talk about some of the actions that describe daily routine or personal care (getting up, taking a shower, getting dressed, etc.), we use reflexive verbs. The pronoun "se" (in blue) attached to an infinitive verb (levantar se , peinar se , despertar se …), indicates that a verb is reflexive.
Conjugation
Reflexive verbs require reflexive pronouns to indicate that the subject of the sentence receives the action of the verb (oneself): me, te, se, nos, os, se. The reflexive pronoun and the conjugation ending must agree. Analyze the conjugations below paying special attention to the reflexive pronouns and conjugation endings in blue.
despertar se (e:ie) vestir se (e:i)
me
despiert
o
nos
despert
amos
me
vist
o
nos
vest
imos
te despiert as os despert áis te vist es os vest ís
se despiert a se despiert an se vist e se vist en
Both the verb ending and the reflexive pronoun agree with the subject.
Yo
me
pong
o
los
zapatos. Vosotros
os
despert
áis
temprano.
I put on my shoes (myself). You (all) wake up early (yourselves).
¡Ojo!: Unlike English, parts of the body or clothing are not used with the possessive pronouns, instead, use definite or indefinite articles instead.
Common reflexive verbs
In the chart below, the "se" ending in blue indicates that the verb is reflexive.
| Verbo | Inglés | Verbo | Inglés |
|---|---|---|---|
| acordar se (de) (o:ue) | to remember | llamar se | to be named |
| acostar se (o:ue) | to go to bed | maquillar se | to put on makeup |
| afeitar se | to shave | peinar se | to comb one's hair |
| bañar se | to bathe; to take a bath | poner se (+noun) | to put on (+item) |
| cepillar se (los dientes, el pelo) | to brush (one's teeth, hair) | poner se (+adj.) | to become (+adj.) |
| despertar se (e:ie) | to wake up | probar se (o:ue) | to try on |
| dormir se (o:ue) | to fall asleep | quedar se | to remain; to stay |
| duchar se | to take a shower | quitar se | to take off (something) |
| enojar se | to get angry | secar se (las manos, el pelo) | to dry (one's hands, one's hair) |
| ir se | to go away; to leave | sentar se (e:ie) | to sit down |
| lavar se (las manos, el pelo) | to wash (one's hands, hair) | sentir se (e:ie) | to feel |
| levantar se | to get up | vestir se (e:i) | to get dressed |
¡Ojo!: Many of the reflexive verbs can be also used as a non-reflexive verb. The meaning will change. Study the following pairs of sentences and the change in meaning created by the addition of the reflexive pronouns. In the example below, non-reflexive verbs are indicated in bold, and the reflexive verbs are in bold with the reflexive pronouns and conjugation endings in blue.


13.3: Database Principles
Learning Outcomes
- Discuss database principles
A database is a shared collection of related data used to support the activities of a particular organization. A database can be viewed as a repository of data that is defined once and then accessed by various users.
A database has the following properties:
- It is a representation of some aspect of the real world or a collection of data elements (facts) representing real-world information.
- A database is logical, coherent and internally consistent.
- A database is designed, built and populated with data for a specific purpose.
- Each data item is stored in a field.
- A combination of fields makes up a table. For example, each field in an employee table contains data about an individual employee. A database can contain many tables. For example, a membership system may contain an address table and an individual member table.
Managing information means taking care of it so that it works for us and is useful for the tasks we perform. By using a DBMS, the information we collect and add to its database is no longer subject to accidental disorganization. It becomes more accessible and integrated with the rest of our work.
Managing information using a database allows us to become strategic users of the data we have. We often need to access and re-sort data for various uses. These may include:
- Creating mailing lists
- Writing management reports
- Generating lists of selected news stories
- Identifying various client needs.
The processing power of a database allows it to manipulate the data it houses, so it can:
- Sort
- Match
- Link
- Aggregate
- Skip fields
- Calculate
- Arrange
Because of the versatility of databases, we find them powering all sorts of projects. A database can be linked to:
- A website that is capturing registered users
- A client-tracking application for social service organizations
- A medical record system for a health care facility
- Your personal address book in your email client
- A collection of word-processed documents
- A system that issues airline reservations
There are a number of characteristics that distinguish the database approach from the file-based system or approach. A database system is referred to as self-describing because it not only contains the database itself, but also metadata which defines and describes the data and relationships between tables in the database. This information is used by the DBMS software or database users if needed. This separation of data and information about the data makes a database system totally different from the traditional file-based system in which the data definition is part of the application programs.
In the file-based system, the structure of the data files is defined in the application programs so if a user wants to change the structure of a file, all the programs that access that file might need to be changed as well. On the other hand, in the database approach, the data structure is stored in the system catalogue and not in the programs. Therefore, one change is all that is needed to change the structure of a file. This insulation between the programs and data is also called program-data independence.
A database supports multiple views of data. A view is a subset of the database, which is defined and dedicated for particular users of the system. Multiple users in the system might have different views of the system. Each view might contain only the data of interest to a user or group of users. By its very nature, a DBMS permits many users to have access to its database either individually or simultaneously. It is not important for users to be aware of how and where the data they access is stored.
Current database systems are designed for multiple users. That is, they allow many users to access the same database at the same time. This access is achieved through features called concurrency control strategies. These strategies ensure that the data accessed are always correct and that data integrity is maintained. The design of modern multiuser database systems is a great improvement from those in the past which restricted usage to one person at a time. Control of data redundancy In the database approach, ideally, each data item is stored in only one place in the database. In some cases, data redundancy still exists to improve system performance, but such redundancy is controlled by application programming and kept to minimum by introducing as little redundancy as possible when designing the database.
The integration of all the data, for an organization, within a database system has many advantages. First, it allows for data sharing among employees and others who have access to the system. Second, it gives users the ability to generate more information from a given amount of data than would be possible without the integration. Enforcement of integrity constraints Database management systems must provide the ability to define and enforce certain constraints to ensure that users enter valid information and maintain data integrity. A database constraint is a restriction or rule that dictates what can be entered or edited in a table such as a postal code using a certain format or adding a valid city in the City field. There are many types of database constraints. Data type, for example, determines the sort of data permitted in a field, for example numbers only. Data uniqueness such as the primary key ensures that no duplicates are entered. Constraints can be simple (field based) or complex (programming).
Not all users of a database system will have the same accessing privileges. For example, one user might have read-only access (i.e., the ability to read a file but not make changes), while another might have read and write privileges, which is the ability to both read and modify a file. For this reason, a database management system should provide a security subsystem to create and control different types of user accounts and restrict unauthorized access.
Another advantage of a database management system is how it allows for data independence. In other words, the system data descriptions or data describing data (metadata) are separated from the application programs. This is possible because changes to the data structure are handled by the database management system and are not embedded in the program itself.
A database management system must include concurrency control subsystems. This feature ensures that data remains consistent and valid during transaction processing even if several users update the same information.
Contributors and Attributions
- Database Design. Authored by : Adrienne Watt. Located at : https://opentextbc.ca/dbdesign01/ . License : CC BY: Attribution


Skip to main content
-
acetylcholine (ACh)
-
major neurotransmitter of the cholinergic system
-
adrenergic receptors
-
mediate responses to epinephrine and norepinephrine; include alpha and beta receptors
-
affinity
-
strength of the attraction between a drug and its receptor
-
Alzheimer’s disease (AD)
-
most common neurodegenerative condition of the brain; characterized by significant changes in brain tissue
-
apoptosis
-
programmed cell death prompted by a signal and designed to replace old cells with new ones
-
autoantibodies
-
produced by the immune system and are directed against one or more of the individual’s own cells, tissues, and proteins instead of foreign invaders
-
cholinergic receptors
-
mediate responses to acetylcholine; include muscarinic and nicotinic receptors
-
chronotropic
-
the increase or decrease of the heart rate
-
delusions
-
false, fixed beliefs not shared by others
-
diplopia
-
double vision—seeing two images of a single object
-
dysarthria
-
difficulty speaking because of muscle weakness
-
dysphagia
-
difficulty swallowing
-
hallucinations
-
perceiving something to be real in the absence of actual stimuli (visual, auditory, olfactory, gustatory, or tactile)
-
intrinsic activity
-
ability of a drug to activate a receptor upon binding
-
microtubules
-
components of a cell skeleton that provide structure and shape to cells, facilitate cell movement and cell division, and transport nutrients/substances within cells
-
myasthenia gravis (MG)
-
a progressive autoimmune neuromuscular disorder characterized by fluctuating muscle weakness and the onset of rapid fatigue
-
neurotoxic
-
drugs that alter the proper functioning of the nervous system
-
norepinephrine
-
neurotransmitter released by almost all of the postganglionic neurons of the SNS
-
parasympathetic nervous system (PSNS)
-
a division of the autonomic nervous system that carries the predominant tone in most organs with the exception of the blood vessels; responsible for the rest and digest functions
-
parasympatholytic
-
drugs that oppose the effects of the parasympathetic nervous system (also known as anticholinergic action)
-
parasympathomimetic
-
medications that activate the parasympathetic nervous system by mimicking the effects of acetylcholine
-
postganglionic neuron
-
neuron that goes from the ganglia to effector organs
-
preganglionic neuron
-
neuron that goes from the spinal cord to the ganglia and releases neurotransmitters
-
ptosis
-
drooping of the upper eyelid
-
remission
-
disappearance of the signs and symptoms of a disease
-
steady state
-
amount of drug eliminated equals the amount of drug within the circulation
-
Stevens-Johnson syndrome
-
hypersensitivity response to certain drugs characterized by lesions of the skin and mucous membranes, fever, malaise, and toxemia
-
sympathetic nervous system (SNS)
-
a division of the autonomic nervous system that is responsible for the fight-or-flight response


10.5E: Exercises
Practice Makes Perfect
In the following exercises, use the properties of logarithms to evaluate.
-
- \(\log _{4} 1\)
- \(\log _{8} 8\)
-
- \(\log _{12} 1\)
- \(\ln e\)
-
- \(3^{\log _{3} 6}\)
- \(\log _{2} 2^{7}\)
-
- \(5^{\log _{5} 10}\)
- \(\log _{4} 4^{10}\)
-
- \(8^{\log _{8} 7}\)
- \(\log _{6} 6^{-2}\)
-
- \(6^{\log _{6} 15}\)
- \(\log _{8} 8^{-4}\)
-
- \(10^{\log \sqrt{5}}\)
- \(\log 10^{-2}\)
-
- \(10^{\log \sqrt{3}}\)
- \(\log 10^{-1}\)
-
- \(e^{\ln 4}\)
- \(\ln e^{2}\)
-
- \(e^{\ln 3}\)
- \(\ln e^{7}\)
- Answer
-
2.
- \(0\)
- \(1\)
4.
- \(10\)
- \(10\)
6.
- \(15\)
- \(-4\)
8.
- \(\sqrt{3}\)
- \(-1\)
10.
- \(3\)
- \(7\)
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
- \(\log _{4} 6 x\)
- \(\log _{5} 8 y\)
- \(\log _{2} 32 x y\)
- \(\log _{3} 81 x y\)
- \(\log 100 x\)
- \(\log 1000 y\)
- Answer
-
2. \(\log _{5} 8+\log _{5} y\)
4. \(4+\log _{3} x+\log _{3} y\)
6. \(3+\log y\)
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
- \(\log _{3} \frac{3}{8}\)
- \(\log _{6} \frac{5}{6}\)
- \(\log _{4} \frac{16}{y}\)
- \(\log _{5} \frac{125}{x}\)
- \(\log \frac{x}{10}\)
- \(\log \frac{10,000}{y}\)
- \(\ln \frac{e^{3}}{3}\)
- \(\ln \frac{e^{4}}{16}\)
- Answer
-
2. \(\log _{6} 5-1\)
4. \(3-\log _{5} x\)
6. \(4-\log y\)
8. \(4-\ln 16\)
In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.
- \(\log _{3} x^{2}\)
- \(\log _{2} x^{5}\)
- \(\log x^{-2}\)
- \(\log x^{-3}\)
- \(\log _{4} \sqrt{x}\)
- \(\log _{5} \sqrt[3]{x}\)
- \(\ln x^{\sqrt{3}}\)
- \(\ln x^{\sqrt[3]{4}}\)
- Answer
-
2. \(5\log _{2} x\)
4. \(-3 \log x\)
6. \(\frac{1}{3} \log _{5} x\)
8. \(\sqrt[3]{4} \ln x\)
In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
- \(\log _{5}\left(4 x^{6} y^{4}\right)\)
- \(\log _{2}\left(3 x^{5} y^{3}\right)\)
- \(\log _{3}\left(\sqrt{2} x^{2}\right)\)
- \(\log _{5}\left(\sqrt[4]{21} y^{3}\right)\)
- \(\log _{3} \frac{x y^{2}}{z^{2}}\)
- \(\log _{5} \frac{4 a b^{3} c^{4}}{d^{2}}\)
- \(\log _{4} \frac{\sqrt{x}}{16 y^{4}}\)
- \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)
- \(\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}\)
- \(\log _{3} \frac{\sqrt{3 x+2 y^{2}}}{5 z^{2}}\)
- \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}\)
- \(\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}\)
- Answer
-
2. \(\log _{2} 3+5 \log _{2} x+3 \log _{2} y\)
4. \(\frac{1}{4} \log _{5} 21+3 \log _{5} y\)
6. \(\begin{array}{l}{\log _{5} 4+\log _{5} a+3 \log _{5} b} {+4 \log _{5} c-2 \log _{5} d}\end{array}\)
8. \(\frac{2}{3} \log _{3} x-3-4 \log _{3} y\)
10. \(\frac{1}{2} \log _{3}\left(3 x+2 y^{2}\right)-\log _{3} 5-2 \log _{3} z\)
12. \(\begin{array}{l}{\frac{1}{3}\left(\log _{5} 3+2 \log _{5} x-\log _{5} 4\right.} {-3 \log _{5} y-\log _{5} z )}\end{array}\)
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
- \(\log _{6} 4+\log _{6} 9\)
- \(\log 4+\log 25\)
- \(\log _{2} 80-\log _{2} 5\)
- \(\log _{3} 36-\log _{3} 4\)
- \(\log _{3} 4+\log _{3}(x+1)\)
- \(\log _{2} 5-\log _{2}(x-1)\)
- \(\log _{7} 3+\log _{7} x-\log _{7} y\)
- \(\log _{5} 2-\log _{5} x-\log _{5} y\)
- \(4 \log _{2} x+6 \log _{2} y\)
- \(6 \log _{3} x+9 \log _{3} y\)
- \(\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)\)
- \(\log \left(x^{2}+2 x+1\right)-2 \log (x+1)\)
- \(4 \log x-2 \log y-3 \log z\)
- \(3 \ln x+4 \ln y-2 \ln z\)
- \(\frac{1}{3} \log x-3 \log (x+1)\)
- \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\)
- Answer
-
2. \(2\)
4. \(2\)
6. \(\log _{2} \frac{5}{x-1}\)
8. \(\log _{5} \frac{2}{x y}\)
10. \(\log _{3} x^{6} y^{9}\)
12. \(0\)
14. \(\ln \frac{x^{3} y^{4}}{z^{2}}\)
16. \(\log (2 x+3)^{2} \cdot \sqrt{x+1}\)
In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.
- \(\log _{3} 42\)
- \(\log _{5} 46\)
- \(\log _{12} 87\)
- \(\log _{15} 93\)
- \(\log _{\sqrt{2}} 17\)
- \(\log _{\sqrt{3}} 21\)
- Answer
-
2. \(2.379\)
4. \(1.674\)
6. \(5.542\)
- Write the Product Property in your own words. Does it apply to each of the following? \(\log _{a} 5 x, \log _{a}(5+x)\). Why or why not?
- Write the Power Property in your own words. Does it apply to each of the following? \(\log _{a} x^{p},\left(\log _{a} x\right)^{r}\). Why or why not?
- Use an example to show that \(\log (a+b) \neq \log a+\log b ?\)
- Explain how to find the value of \(\log _{7} 15\) using your calculator.
- Answer
-
2. Answers may vary
4. Answers may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. On a scale of \(1−10\), how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


2.4: Interviewing
- Explain the various types of interviews and interview questions.
- Discuss interview methods and potential mistakes in interviewing candidates.
- Explain the interview process.
Interviewing people costs money. As a result, after candidates are selected, good use of time is critical to making sure the interview process allows for selection of the right candidate. In an unstructured interview, questions are changed to match the specific applicant; for example, questions about the candidate’s background in relation to their résumé might be used. In a structured interview, there is a set of standardized questions based on the job analysis, not on individual candidates’ résumés. While a structured interview might seem the best option to find out about a particular candidate, the bigger concern is that the interview revolves around the specific job for which the candidate is interviewing. In a structured interview, the expected or desired answers are determined ahead of time, which allows the interviewer to rate responses as the candidate provides answers. This allows for a fair interview process, according to the US Office of Personnel Management 1 . For purposes of this section, we will assume that all interviews you perform will be structured, unless otherwise noted.
Types of Interviews
Interview processes can be time-consuming, so it makes sense to choose the right type of interview(s) for the individual job. Some jobs, for example, may necessitate only one interview, while another may necessitate a telephone interview and at least one or two traditional interviews. Keep in mind, though, that there will likely be other methods with which to evaluate a candidate’s potential, such as testing. Testing is discussed in Section “Testing”. Here are different types of interviews:
- Traditional interview. This type of interview normally takes place in the office. It consists of the interviewer and the candidate, and a series of questions are asked and answered.
- Telephone interview. A telephone interview is often used to narrow the list of people receiving a traditional interview. It can be used to determine salary requirements or other data that might automatically rule out giving someone a traditional interview. For example, if you receive two hundred résumés and narrow these down to twenty-five, it is still unrealistic to interview twenty-five people in person. At this point, you may decide to conduct phone interviews of those twenty-five, which could narrow the in-person interviews to a more manageable ten or so people.
- Panel interview. A panel interview occurs when several people are interviewing one candidate at the same time. While this type of interview can be nerve racking for the candidate, it can also be a more effective use of time. Consider some companies who require three to four people to interview candidates for a job. It would be unrealistic to ask the candidate to come in for three or four interviews, so it makes sense for them to be interviewed by everyone at once.
- Information interview. Informational interviews are usually used when there is no specific job opening, but the candidate is exploring possibilities in a given career field. The advantage to conducting these types of interviews is the ability to find great people ahead of a job opening.
- Meal interviews. Many organizations offer to take the candidate to lunch or dinner for the interview. This can allow for a more casual meeting where, as the interviewer, you might be able to gather more information about the person, such as their manners and treatment of waitstaff. This type of interview might be considered an unstructured interview, since it would tend to be more of a conversation as opposed to a session consisting of specific questions and answers.
- Group interview. In a group interview, two or more candidates interview at the same time. This type of interview can be an excellent source of information if you need to know how they may relate to other people in their job.
- Video interviews. Video interviews are the same as traditional interviews, except that video technology is used. This can be cost saving if one or more of your candidates are from out of town. Skype, for example, allows free video calls. An interview may not feel the same as a traditional interview, but the same information can be gathered about the candidate.
- Nondirective interview (sometimes called an unstructured interview). In a nondirective interview, the candidate essentially leads the discussion. Some very general questions that are planned ahead of time may be asked, but the candidate spends more time talking than the interviewer. The questions may be more open ended; for example, instead of asking, “Do you like working with customers?” you may ask, “What did you like best about your last job?” The advantage of this type of interview is that it can give candidates a good chance to show their abilities; however, the downside is that it may be hard to compare potential candidates, since questions are not set in advance. It relies on more of a “gut feeling” approach.
It is likely you may use one or more of these types of interviews. For example, you may conduct phone interviews, then do a meal interview, and follow up with a traditional interview, depending on the type of job.
Interview Questions
Most interviews consist of many types of questions, but they usually lean toward situational interviews or behavior description interviews. A situational interview is one in which the candidate is given a sample situation and is asked how he or she might deal with the situation. In a behavior description interview, the candidate is asked questions about what he or she actually did in a variety of given situations. The assumption in this type of interview is that someone’s past experience or actions are an indicator of future behavior. These types of questions, as opposed to the old “tell me about yourself” questions, tend to assist the interviewer in knowing how a person would handle or has handled situations. These interview styles also use a structured method and provide a better basis for decision making. Examples of situational interview questions might include the following:
- If you saw someone stealing from the company, what would you do?
- One of your employees is performing poorly, but you know he has some personal home issues he is dealing with. How would you handle complaints from his colleagues about lack of performance?
 What would you do?
- You are rolling out a new sales plan on Tuesday, which is really important to ensure success in your organization. When you present it, the team is lukewarm on the plan. What would you do?
- You disagree with your supervisor on her handling of a situation. What would you do?
Examples of behavior description interview questions might include the following:
- Tell me about a time you had to make a hard decision. How did you handle this process?
- Give an example of how you handled an angry customer.
- Do you show leadership in your current or past job? What would be an example of a situation in which you did this?
- What accomplishments have given you the most pride and why?
- What plans have you made to achieve your career goals?
Top 36 Interview Questions and Answers
" href=" http://www.youtube.com/watch?v=3L3V5hg4QDE " class="replaced-iframe">(click to see video)
Examples of how to answer those difficult interview questions.
As you already know, there are many types of interview questions that would be considered illegal. Here are some examples:
- National origin. You cannot ask seemingly innocent questions such as “That’s a beautiful name, where is your family from?” This could indicate national origin, which could result in bias. You also cannot ask questions about citizenship, except by asking if a candidate is legally allowed to work in the United States. Questions about the first language of the candidate shouldn’t be asked, either. However, asking “Do you have any language abilities that would be helpful in this job?” or “Are you authorized to work in the United States?” would be acceptable.
- Age. You cannot ask someone how old they are, and it is best to avoid questions that might indicate age, such as “When did you graduate from high school?” However, asking “Are you over 18?” is acceptable.
- Marital status. You can’t ask direct questions about marital status or ages of children. An alternative may be to ask, “Do you have any restrictions on your ability to travel, since this job requires 50 percent travel?”
- Religion. It’s illegal to ask candidates about their religious affiliation or to ask questions that may indicate a religion-affiliated school or university.
- Disabilities. You may not directly ask if the person has disabilities or recent illnesses. You can ask if the candidate is able to perform the functions of the job with or without reasonable accommodations.
- Criminal record. While it is fine to perform a criminal record check, asking a candidate if they have ever been arrested is not appropriate; however, questions about convictions and guilty pleadings are acceptable.
- Personal questions. Avoid asking personal questions, such as questions about social organizations or clubs, unless they relate to the job.
Besides these questions, any specific questions about weight, height, gender, and arrest record (as opposed to allowable questions about criminal convictions) should be avoided.
HR professionals and managers should be aware of their own body language in an interview. Some habits, such as nodding, can make the candidate think they are on the right track when answering a question. Also, be aware of a halo effect or reverse halo effect. This occurs when an interviewer becomes biased because of one positive or negative trait a candidate possesses. Interview bias can occur in almost any interview situation. Interview bias is when an interviewer makes assumptions about the candidate that may not be accurate (Lipschultz, 2010). These assumptions can be detrimental to an interview process. Contrast bias is a type of bias that occurs when comparing one candidate to others. It can result in one person looking particularly strong in an area, when in fact they look strong compared to the other candidates. A gut feeling bias is when an interviewer relies on an intuitive feeling about a candidate. Generalization bias can occur when an interviewer assumes that how someone behaves in an interview is how they always behave. For example, if a candidate is very nervous and stutters while talking, an assumption may be made that he or she always stutters. Another important bias called cultural noise bias occurs when a candidate thinks he or she knows what the interviewer wants to hear and answers the questions based on that assumption. Nonverbal behavior bias occurs when an interviewer likes an answer and smiles and nods, sending the wrong signal to the candidate. A similar to me bias (which could be considered discriminatory) results when an interviewer has a preference for a candidate because he or she views that person as having similar attributes as themselves. Finally, recency bias occurs when the interviewer remembers candidates interviewed most recently more so than the other candidates.
What are the dangers of a reverse halo effect?
Interview Process
Once the criteria have been selected and interview questions developed, it is time to start interviewing people. Your interviewing plan can determine the direction and process that should be followed:
- Recruit new candidates.
- Establish criteria for which candidates will be rated.
- Develop interview questions based on the analysis.
- Set a time line for interviewing and decision making.
- Connect schedules with others involved in the interview process.
- Set up the interviews with candidates and set up any testing procedures.
- Interview the candidates and perform any necessary testing.
- Once all results are back, meet with the hiring team to discuss each candidate and make a decision based on the established criteria.
- Put together an offer for the candidate.
As you can see, a large part of the interviewing process is planning. For example, consider the hiring manager who doesn’t know exactly the type of person and skills she is looking to hire but sets up interviews anyway. It is difficult, if not impossible, to determine who should be hired if you don’t know what you are looking for in the first place. In addition, utilizing time lines for interviewing can help keep everyone involved on track and ensure the chosen candidate starts work in a timely manner. Here are some tips to consider when working with the interview process:
- Make sure everyone is trained on the interviewing process. Allowing someone who has poor interviewing skills to conduct the interview will likely not result in the best candidate. In a worst-case scenario, someone could ask an illegal question, and once hired, the candidate can sue the organization. UCLA researchers (Hanricks, 2011) calculated that plaintiffs win about half of hiring discrimination cases that go to trial, sometimes because of interviewers asking illegal questions. For example, “I see you speak Spanish, where did you study it?” is a seemingly harmless question that could be indirectly asking a candidate his or her ethnic background. To avoid such issues, it’s important to train managers in the proper interviewing process.
- Listen to the candidate and try to develop a rapport with them. Understand how nervous they must be and try to put them at ease.
- Be realistic about the job. Do not try to paint a “rosy” picture of all aspects of the job. Being honest up front helps a candidate know exactly what they will be in for when they begin their job.
- Be aware of your own stereotypes and do not let them affect how you view a potential candidate.
- Watch your own body language during the interview and that of the candidate. Body language is a powerful tool in seeing if someone is the right fit for a job. For example, Scott Simmons, vice president at Crist|Kolder, interviewed someone for a CFO position. The candidate had a great résumé, but during the interview, he offered a dead-fish handshake, slouched, and fidgeted in his chair. The candidate didn’t make eye contact and mumbled responses, and, of course, he didn’t get the job (Reeves, 2006), because his body language did not portray the expectations for the job position.
- Stick to your criteria for hiring. Do not ask questions that have not been predetermined in your criteria.
- Learn to manage disagreement and determine a fair process if not everyone on the interviewing team agrees on who should be hired. Handling these types of disagreements is discussed further in Chapter 9 “Successful Employee Communication”.
Once you have successfully managed the interview process, it is time to make the decision. Section 5.4.1 “Testing” discusses some of the tools we can use to determine the best candidate for the job.
Can you think of a time when the interviewer was not properly trained? What were the results?
Silly Job Interview— Monty Python
" href=" http://www.youtube.com/watch?v=zP0sqRMzkwo " class="replaced-iframe">(click to see video)
An exaggerated and funny example of an untrained interviewer.
- Traditional, telephone, panel, informational, meal, group, and video are types of interviews. A combination of several of these may be used to determine the best candidate for the job. A structured interview format means the questions are determined ahead of time, and unstructured means the questions are based on the individual applicant. The advantage of a structured interview is that all candidates are rated on the same criteria. Before interviewing occurs, criteria and questions for a structured interview should be developed.
-
Interview questions can revolve around
situational questions
or
behavioral questions
. Situational questions focus on asking someone what they would do in a given situation, while behavioral questions ask candidates what they have done in certain situations.
- Interview questions about national origin, marital status, age, religion, and disabilities are illegal. To avoid any legal issues, it is important for interviewers to be trained on which questions cannot be asked. The halo effect , which assumes that one desirable trait means all traits are desirable, should also be avoided.
- The process involved in interviewing a person includes the following steps: recruit new candidates; establish criteria for which candidates will be rated; develop interview questions based on the analysis; set a time line for interviewing and decision making; connect schedules with others involved in the interview process; set up interviews with candidates and set up any testing procedures; interview the candidates and perform any necessary testing; and once all results are back, meet with the hiring team to discuss each candidate and make a decision based on the established criteria; then finally, put together an offer for the candidate.
- Developing a rapport, being honest, and managing the interview process are tips to having a successful interview.
- With a partner, develop a list of five examples (not already given in the chapter) of situational and behavioral interview questions.
- Why is it important to determine criteria and interview questions before bringing someone in for an interview?
- Visit Monster.com and find two examples of job postings that ask those with criminal records not to apply. Do you think, given the type of job, this is a reasonable criteria?
1 “Structured Interviews: A Practical Guide,” US Office of Personnel Management, September 2008, accessed January 25, 2011, https://apps.opm.gov/ADT/ContentFiles/SIGuide09.08.08.pdf .
References
Hanricks, M., “3 Interview Questions That Could Cost You $1 Million,” BNET , March 8, 2011, accessed August 2, 2011, http://www.bnet.com/blog/business-myths/3-interview-questions-that-could-cost-your-company-1-million/791 .
Lipschultz, J., “Don’t Be a Victim of Interview Bias,” Career Builder , June 15, 2010, accessed July 12, 2011, jobs.aol.com/articles/2010/06/15/interview-bias/.
Reeves, S., “Is Your Body Betraying You in Job Interviews?” Forbes , February 2006, accessed August 2, 2011, http://www.forbes.com/2006/02/15/employment-careers-interviews-cx_sr_0216bizbasics.html .


11.2: The Nuclear Equation
In Chapter 1 , we described the meaning of the atomic symbol for an element. In the atomic symbol, the atomic number (the number of protons in the nucleus) appears as a subscript preceding the symbol for the element. The mass number appears as a superscript, also preceding the symbol. Thus for uranium (atomic number 92) with a mass of 238, the symbol is \(\ce{_{92}^{238}U}\). To show radioactive decay in a chemical equation, you need to use atomic symbols. Thus, for the loss of an alpha particle from \(\ce{ _{92}^{238}U}\), you need to show uranium on the “reactant” side of the equation and thorium and the alpha particle on the “product” side. Just like any other chemical equation, a nuclear equation must balance . The sum of the mass numbers on the reactant side must equal the sum of the mass numbers on the product side. Because we started with uranium-238 and lost four mass units in the alpha particle, the product (or products) of the decay must have a total mass of (238 – 4) = 234 . We have also removed two protons from the uranium nucleus, dropping the atomic number by two. The newly formed element is therefore thorium-234.
\[_{92}^{238}\textrm{U}\rightarrow _{2}^{4}\textrm{He}+_{90}^{234}\textrm{Th} \nonumber \]
In this equation, we have shown the alpha particle using the atomic symbol for helium (\(\ce{_2^4 He}\)), but this is often shown using the symbol \(\ce{_{2}^{4}\alpha}\) . Compounds that emit alpha particles are very toxic, in spite of the poor penetrating ability of the particle. This is especially true if the emitting element is inhaled or ingested. The toxic dose of the alpha-emitter \(\ce{^{210}Po}\) in a 175-pound person has been estimated to be about one microgram (1 × 10 -6 g).
Thorium-230 and polonium-210 both undergo loss of an alpha particle to form different elements. For each of these radioactive decay processes, write the appropriate nuclear equation and show the nature of the elements that are formed.
Radium-226 and polonium-214 both undergo loss of an alpha particle to form different elements. For each of these radioactive decay processes, write the appropriate nuclear equation and show the nature of the elements that are formed.


13.34: Outcome- Defining the Message
What you’ll learn to do: discuss how to develop effective messaging for marketing communications
At the center of any successful marketing activity is a message. Without a solid, consistent message, your marketing efforts are like a compass without an arrow: there is nothing to point your target audiences in the direction you want them to go.
Good messaging takes time and attention to develop, but this effort pays a huge dividend down the road—when your marketing activities have their desired effect on the hearts, minds, and wallets of the people you want to reach.
The specific things you’ll learn in this section include:
- Explain the role of consistent messaging in strengthening the impact of marketing communications
- Outline a standard framework for developing messaging for marketing communications
- Explain the importance of including a clear call to action in marketing communications
Learning Activities
- Reading: Defining the Message
- Self Check: Defining the Message
Contributors and Attributions
CC licensed content, Original
- Outcome: Defining the Message. Provided by : Lumen Learning. License : CC BY: Attribution


1.2.1: Scale Drawings
Lesson
Let's explore scale drawings.
Exercise \(\PageIndex{1}\): What is a Scale Drawing?
Here are some drawings of a school bus, a quarter, and the subway lines around Boston, Massachusetts. The first three drawings are scale drawings of these objects.
The next three drawings are not scale drawings of these objects.
Discuss with your partner what a scale drawing is.
Exercise \(\PageIndex{2}\): Sizing Up a Basketball Court
Your teacher will give you a scale drawing of a basketball court. The drawing does not have any measurements labeled, but it says that 1 centimeter represents 2 meters.
- Measure the distances on the scale drawing that are labeled a–d to the nearest tenth of a centimeter. Record your results in the first row of the table.
- The statement “1 cm represents 2 m” is the scale of the drawing. It can also be expressed as “1 cm to 2 m,” or “1 cm for every 2 m.” What do you think the scale tells us?
-
How long would each measurement from the first question be on an actual basketball court? Explain or show your reasoning.
measurement (a) length of court (b) width of court (c) hoop to hoop (d) 3 point line to sideline scale drawing actual court Table \(\PageIndex{1}\) -
On an actual basketball court, the bench area is typically 9 meters long.
- Without measuring, determine how long the bench area should be on the scale drawing.
- Check your answer by measuring the bench area on the scale drawing. Did your prediction match your measurement?
Exercise \(\PageIndex{3}\): Tall Structures
Here is a scale drawing of some of the world’s tallest structures.
- About how tall is the actual Willis Tower? About how tall is the actual Great Pyramid? Be prepared to explain your reasoning.
- About how much taller is the Burj Khalifa than the Eiffel Tower? Explain or show your reasoning.
- Measure the line segment that shows the scale to the nearest tenth of a centimeter. Express the scale of the drawing using numbers and words.
Are you ready for more?
The tallest mountain in the United States, Mount Denali in Alaska, is about 6,190 m tall. If this mountain were shown on the scale drawing, how would its height compare to the heights of the structures? Explain or show your reasoning.
Summary
Scale drawings are two-dimensional representations of actual objects or places. Floor plans and maps are some examples of scale drawings. On a scale drawing:
- Every part corresponds to something in the actual object.
- Lengths on the drawing are enlarged or reduced by the same scale factor.
- A scale tells us how actual measurements are represented on the drawing. For example, if a map has a scale of “1 inch to 5 miles” then a \(\frac{1}{2}\)-inch line segment on that map would represent an actual distance of 2.5 miles
Sometimes the scale is shown as a segment on the drawing itself. For example, here is a scale drawing of a stop sign with a line segment that represents 25 cm of actual length.
The width of the octagon in the drawing is about three times the length of this segment, so the actual width of the sign is about \(3\cdot 25\), or 75 cm.
Because a scale drawing is two-dimensional, some aspects of the three-dimensional object are not represented. For example, this scale drawing does not show the thickness of the stop sign.
A scale drawing may not show every detail of the actual object; however, the features that are shown correspond to the actual object and follow the specified scale.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
The Westland Lysander was an aircraft used by the Royal Air Force in the 1930s. Here are some scale drawings that show the top, side, and front views of the Lysander.
Use the scales and scale drawings to approximate the actual lengths of:
- the wingspan of the plane, to the nearest foot
- the height of the plane, to the nearest foot
- the length of the Lysander Mk. I, to the nearest meter
Exercise \(\PageIndex{5}\)
A blueprint for a building includes a rectangular room that measures 3 inches long and 5.5 inches wide. The scale for the blueprint says that 1 inch on the blueprint is equivalent to 10 feet in the actual building. What are the dimensions of this rectangular room in the actual building?
Exercise \(\PageIndex{6}\)
Here is a scale map of Lafayette Square, a rectangular garden north of the White House.
- The scale is shown in the lower right corner. Find the actual side lengths of Lafayette Square in feet.
- Use an inch ruler to measure the line segment of the graphic scale. About how many feet does one inch represent on this map?
Exercise \(\PageIndex{7}\)
Here is Triangle A. Lin created a scaled copy of Triangle A with an area of 72 square units.
- How many times larger is the area of of the scaled copy compared to that of Triangle A?
- What scale factor did Lin apply to the Triangle A to create the copy?
- What is the length of bottom side of the scaled copy?
(From Unit 1.1.6)


Skip to main content
-
-
5.1: Axioms of the Real Numbers
-
In this chapter we will take a deep dive into structure of the real numbers by building up the multitude of properties you are familiar with by starting with a collection of fundamental axioms. Recall that an axiom is a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved.
-
-
5.2: Standard Topology of the Real Line
-
n this section, we will introduce the notions of open, closed, compact, and connected as they pertain to subsets of the real numbers. These properties form the underpinnings of a branch of mathematics called topology (derived from the Greek words tópos, meaning ‘place, location’, and ology, meaning ‘study of’). Topology, sometimes called “rubber sheet geometry," is concerned with properties of spaces that are invariant under any continuous deformation.


2.1: Definitions
Let \(X\) and \(Y\) be sets. A relation from \(X\) to \(Y\) is a subset of \(X \times Y\) .
Alternatively, any set of ordered pairs is a relation. If \(Y=X\) , we say that \(R\) is a relation on \(X\) .
Notation. \(x\) Ry Let \(X\) and \(Y\) be sets and \(R\) be a relation on \(X \times Y\) . If \(x \in X\) and \(y \in Y\) , then we may express that \(x\) bears relation \(R\) to \(y\) (that is \((x, y) \in R\) ) by writing \(x R y\) .
So for \(X\) and \(Y\) sets, \(x \in X, y \in Y\) , and \(R\) a relation on \(X \times Y\) , \(x R y\) if and only if \((x, y) \in R\) .
Let \(\leq\) be the usual ordering on \(\mathbb{Q}\) . Then \(\leq\) is a relation on \(\mathbb{Q}\) . We write \[1 / 2 \leq 2\] to express that \(1 / 2\) bears the relation \(\leq\) to 2 .
Define a relation \(R\) from \(\mathbb{Z}\) to \(\mathbb{R}\) by \(x R y\) if \(x>y+3\) . Then we could write \(7 R \sqrt{2}\) or \((7, \sqrt{2}) \in R\) to say that \((7, \sqrt{2})\) is in the relation.
Let \(X=\{2,7,17,27,35,72\}\) . Define a relation \(R\) by \(x R y\) if \(x \neq y\) and \(x\) and \(y\) have a digit in common. Then \(R=\{(2,27),(2,72),(7,17),(7,27),(7,72),(17,7),(17,27),(17,72),\) , \((27,2),(27,7),(27,17),(27,72),(72,2),(72,7),(72,17),(72,27)\}\) . ExAMPLE 2.4. Let \(P\) be the set of all polygons in the plane. Define a relation \(E\) by saying \((x, y) \in E\) if \(x\) and \(y\) have the same number of sides.
How do mathematicians use relations? A relation on a set can be used to impose structure. In Example 2.1, the usual ordering relation \(\leq\) on \(\mathbb{Q}\) allows us to think of rational numbers as lying on a number line, which provides additional insight into rational numbers. In Example 2.4, we can use the relation to break polygons up into the sets of triangles, quadrilaterals, pentagons, etc.
A function \(f: X \rightarrow Y\) can be thought of as a very special sort of relation from \(X\) to \(Y\) . Indeed, the graph of the function is a set of ordered pairs in \(X \times Y\) , but it has the additional property that every \(x\) in \(X\) occurs exactly once as a first element of a pair in the relation. As we discussed in Section 1.3, functions are a useful way to relate sets.
Let \(X\) be a set, and \(R\) a relation on \(X\) . Here are some important properties the relation may or may not have.
\(R\) is reflexive if for every \(x \in X\) , \[x R x .\] Symmetric \(R\) is symmetric if for any \(x, y \in X\) , \[x R y \text { implies } y R x \text {. }\] Antisymmetric \(R\) is antisymmetric if for any \(x, y \in X\) , \[[(x, y) \in R \text { and }(y, x) \in R] \text { implies } x=y \text {. }\] Transitive \(R\) is transitive if for any \(x, y, z \in X\) , \[[x R y \text { and } y R z] \text { implies }[x R z] \text {. }\] Which of these four properties apply to the relations given in Examples 2.1-2.4 (Exercise 2.1)?


8.5: Stability of the Allyl Radical - Resonance Revisited
-
- Last updated
- Save as PDF
Objectives
After completing this section, you should be able to
- explain the stability of the allyl radical in terms of resonance.
- explain the difference between resonance and tautomerism .
- write an equation for the reaction of an unsymmetrical alkene with N-bromosuccinimide.
- draw the structure of each of the possible products that could be obtained from the reaction of a given unsymmetrical alkene with N-bromosuccinimide, and predict which product will predominate.
- explain the formation of more than one product from the reaction of N-bromosuccinimide with a given unsymmetrical alkene.
- explain the observed product ratio when a given unsymmetrical alkene is treated with N-bromosuccinimide.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- delocalized
- resonance forms
- resonance hybrid
Study Notes
You will have encountered the concept of resonance if you have taken general first-year chemistry course. You should also briefly review Section 2.5 .
When we can represent a species by two or more different Lewis or Kekulé structures, neither of which represents the true structure of the species, these structures are referred to as resonance forms . A common example used in general chemistry courses to illustrate the concept of resonance is ozone, O 3 . The two resonance forms of ozone may be represented as follows:
The concept of resonance is quite important, and will be used frequently throughout the remainder of this course. The guidelines below may assist you in drawing resonance contributors.
- Resonance occurs whenever a molecule, radical or ion can be represented by two or more structures differing only in the arrangement of electrons (no atoms may be moved).
- The true structure of a species is a hybrid of the resonance contributors and is more stable (i.e., lower in energy) than any of the contributors.
- The most important contributors are those containing the most covalent bonds. Another way of saying the same thing is that the most important contributors have the least amount of charge separation.
- Contributors in which all the atoms (except hydrogen) have a complete octet (i.e., are surrounded by eight electrons) are particularly important.
In the previous section we discussed the allylic bromination of a symmetrical alkene with NBS such as this cyclopentene, which affords one product.
However, with an unsymmetrical alkene and the delocalized unpaired electron forming various allylic resonances, several products are possible. For example, the NBS bromination of 4-methyl-cyclohexene leads to three products.
The geometry and relative stability of carbon radicals
As organic chemists, we are particularly interested in radical intermediates in which the unpaired electron resides on a carbon atom. Experimental evidence indicates that the three bonds in a carbon radical have trigonal planar geometry, and therefore the carbon is considered to be sp 2 -hybridized with the unpaired electron occupying the perpendicular, unhybridized 2p z orbital. Contrast this picture with carbocation and carbanion intermediates, which are both also trigonal planar but whose 2p z orbitals contain zero or two electrons, respectively.
The trend in the stability of carbon radicals parallels that of carbocations (section 8.4B ): tertiary radicals, for example, are more stable than secondary radicals, followed by primary and methyl radicals. This should make intuitive sense, because radicals, like carbocations, can be considered to be electron deficient, and thus are stabilized by the electron-donating effects of nearby alkyl groups.
Benzylic and allylic radicals are more stable than alkyl radicals due to resonance effects - an unpaired electron can be delocalized over a system of conjugated pi bonds. An allylic radical, for example, can be pictured as a system of three parallel 2p z orbitals sharing three electrons. With two resonance forms, the allylic radical is electronically symmetrical. Due to resonance hybrid theory, neither structure is correct, but instead the structure lies somewhere between the two resonance forms. Another way to phrase this is that the unpaired electron is delocalized across all the carbon atoms through the pi system and not localized on one site. The more resonance structures, the more stable the molecule. This is why the allylic radical is more stable than the alkyl radical.
Because the allylic radical is symmetrical, a reaction can occur on either side. Therefore if reacting with bromine, the bromination could occur on either end of the allylic radical. When the allyl radical is symmetrical, this yields the same product. However, if you have an unsymmetrical allyl radical, it would lead to a mixture of products and not necessarily in equal amounts. This is because the intermediate radical is unsymmetrical. The reaction will occur at the less hindered site. An example would be 1-octene as a starting material in a bromination. The products the reaction would yield would be 3-bromo-1-octene and 1-bromo-2-octene.
Further Reactions
The products from allylic bromination reactions can easily be converted into dienes by elimination using a base. If the alkyl chain is long enough, alkenes can be converted to dienes through a two-step process: allylic bromination followed by elimination.
Exercise \(\PageIndex{1}\)
1) The following reaction shows the major product. Explain why this would be the final product and why the 2° bromo product is not the major product.
2) Predict the products of the following reactions:
- Answer
-
1) The product (A) is a 1° halogen which is more predominant product even though the (B) had a better transition state with a 2° radical. The 1° radical intermediate is not as sterically hindered.
2)