Means and Variances of Linear Combinations of Random Variables Stats Assignment Homework Help

4.3   Means and Variances of Linear Combinations of Random Variables

We now develop some useful properties that will simplify the calculations of means and variances of random variables that appear in later chapters. These properties will permit us to deal with expectations in terms •£ other parameters that are either known or are easily computed. All the results that we present here are .valid for both discrete and continuous random variables^ Proofs are given only for the continuous case. We begin with a theorem and two corollaries that should be. intuitively, reasonable to the reader.

Theorem 4.5:

If a and b are constants, then

Proof: By the definition of an expected value,

The first integral on the right is E(X) and the second integral equals 1. Therefore, we have

y 4.1;   Setting o = 0, yre see that E(b) =6.

4.2:   Setting b = 0, we see that E(aX) » aE(X).

Example 4.15:1 Applying Theorem 4-5 to the discrete random variable f(X) 2X — 1, rework


Corollary 4.10

If X! are independent random variables then

Example 4.20:

If A’ and Y are random variables with v,iriances a\ = 2, ct£ = 4, and covariance (by Theorem 4.9) (by Theorem 4.10) find the variance of the random variable Z = 3X – 4K + 8.

Example 4.21:

Let X and Y .denote the amount of two different types of impurities in a batch of a certain chemical product.  Suppose that X and Y are independent random variables with variances a\ = 2 and a\ = 3.  Find the variance of the random variable Z – 3X – 1Y -f 5.

What if the Function Is Nonlinear?

In that which has preceded this section, we deal with properties of linear functions of random variables for very important reasons. From, Chapter 8 through 15, much of what is discussed and illustrated are practical and real world problems in which the analyst is constructing a linear model to describe a data set and thus to describe or explain the behavior of a certain scientific phenomenon. Thus; it is natural that expected values and variances oflinear combinations of random variables are encountered. However, there are situations in which properties of nonlinear functions of random variables become important. Certainly there are many scientific phenomena that are nonlinear and certainly statistical modeling using nonlinear functions is very important. In fact, in Chapter 12, we deal with the modeling of what have become certain standard nonlinear models. Indeeld, even a simple function of random variables, say Z — X/Y, occurs quite frequently in practice and yet unlike the rules given earlier in this section for the expected value of linear combinations of random variables, there is no simple general rule.

For example,except in very special circumstances. 

The material provided by Theorem 4,5 through Theorem 4.10 and the various corollaries are extremely useful in that there are no restrictions on the form of the density or probability functions, apart from the property of independence when it is required as in the corollaries following Theorem 4.10. To illustrate, consider Example 4.21; the variance of Z = 3X – 2Y + 5 does not require restrictions on the distributions of the amounts X and Y of the two types of impurities. Only independence between X and Y is required. Now, we do 1»ave at our disposal the capacity to-find /9(x) and ff(x, for any function g(-) from first principles established in Theorem 4.1 and 4.3, where it is assumed that the corresponding distribution /(i) is known. Exercises 4.40,4.41, and 4.42, among others, illustrate the use of these theorems. Thus, if the function 3(1) is nonlinear and the density function (or probability function in the discrete case) is known) and can be evaluated exactly. But, Bke the rule given for linear combinations, are there, rules for nonlinear functions that can be used when the form of the distribution of | the pertinent random variables is not known? In general, suppose X is a random variable and Y = g(x). The general solu­tion for E(Y) or Var(Y) can be difficult and depends on the complexity of the j function <?(•). However, there are approximations available that depend on a linear approximation of the function g(x). For. example, suppose we denote E(X] as and Var(X) = crx. Then a Taylor series approximation of 5(1) around I gives As a result, if we.truricate with the linear term and take the expected value of botb sides, we obtain I?[0(.X)] « ff(A*x)> which is certainly intuitive and in some c gives a reasonable approximation. However, if we include the second-order ten of the Tayler series, then we have a second order adjustment for this first-or approximation as

Example 4.2:

Given the random variable X with mean p,x and variance crj, give the second-orde apprcximatioa to Similarly, we can develop an approximation for Var\g(x)} by taking the’ of both sides of the first order Taylor series expansion of g(x). Approximation of Var[g(x)]

Example 4.23:  Given the random variable X as in Example 4.22, give an approximate formula !

These approximations can be extended to nonlinear functions of more than random variable. Given a set of independent random variables X\, Xy,.. and variances <r?, 0f,…, <t£, respectively, let be a nonlinear function; then the following are approximations for E(Y,)

Example 4.24:

Consider two independent random variables, X and Z, with means nx, Hz, and
variances a\ and er|, respectively. Consider a random variable …… Give approximations for E(Y) and Var(Y), Solution: For £(F), we must use  As a result  and the approximation fof the variance of Y is given by

Posted on July 3, 2014 in Statistics

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