<s>[INST] Conditional vs unconditional expectation I'm having trouble understanding the calculation of conditional versus unconditional expectations in this case:
\begin{array}{c c c c}
& \quad    &X=1\quad   &X=-1\quad  &X=2  \\
&Y=1\quad  &0.25\quad  &0.25\quad  &0    \\
&Y=2\quad  &0\quad     &0\quad     &0.5
\end{array}
\begin{align}
~  \\
~  \\
~  \\
E(Y)&=\sum_Y y f(y)  \\
~  \\
~  \\
~  \\
E(Y|X)&=\sum_Y y f(y|x)
\end{align}
To me, both calculations are $1*0.25 + 1*0.25 + 2*0.5 = 1.5$. What am I doing wrong?
 [/INST] If $p_{X,Y}(x,y)$ denotes the joint probability mass function of discrete
random variables $X$ and $Y$, then the marginal mass functions are
$$\begin{align}
p_X(x) &= \sum_y p_{X,Y}(x,y)\\
p_Y(y) &= \sum_x p_{X,Y}(x,y)
\end{align}$$
and so we have that
$$E[Y] = \sum_y y\cdot p_{Y}(y) = \sum_y y\cdot \sum_xp_{X,Y}(x,y)
= \sum_x\sum_y y\cdot p_{X,Y}(x,y).\tag{1}$$
Now, the conditional probability mass function of $Y$ given that 
$X = x$ is
$$p_{Y\mid X}(y \mid X=x) = \frac{p_{X,Y}(x,y)}{p_X(x)}
= \frac{p_{X,Y}(x,y)}{\sum_y p_{X,Y}(x,y)}\tag{2}$$
and
$$E[Y\mid X=x] = \sum_y y\cdot p_{Y\mid X}(y \mid X=x).\tag{3}$$
The value of this expectation depends on our choice of the value $x$
taken on by $X$ and is thus a random variable; indeed, it is a function
of the random variable $X$, and this random variable is denoted
$E[Y\mid X]$. It happens to take on values 
$E[Y\mid X = x_1], E[Y\mid X=x_2], \cdots $ with probabilities
$p_X(x_1), p_X(x_2), \cdots$ and so its expected value is
$$\begin{align}E\bigr[E[Y\mid X]\bigr] &= \sum_x E[Y\mid X = x]\cdot p_X(x)
&\text{note the sum is w.r.t}~x\\
&=  \sum_x \left[\sum_y y\cdot p_{Y\mid X}(y \mid X=x)\right]\cdot p_X(x)
&\text{using}~ (3)\\
&= \sum_x \left[\sum_y y\cdot \frac{p_{X,Y}(x,y)}{p_X(x)}\right]\cdot p_X(x)
&\text{using}~ (2)\\
&= \sum_x \sum_y y\cdot p_{X,Y}(x,y)\\
&= E[Y] &\text{using}~(1)
\end{align}$$

In general, the number $E[Y\mid X = x]$ need not equal
the number $E[Y]$ for any $x$.  But, if $X$ and $Y$ are
independent random variables and so $p_{X,Y}(x,y) = p_X(x)p_Y(y)$
for all $x$ and $y$, then
$$p_{Y\mid X}(y \mid X=x) = \frac{p_{X,Y}(x,y)}{p_X(x)}
= \frac{p_X(x)p_Y(y)}{p_X(x)} = p_Y(y)\tag{4}$$
and so $(3)$ gives
$$E[Y\mid X=x] = \sum_y y\cdot p_{Y\mid X}(y \mid X=x)
= \sum_y y\cdot p_Y(y) = E[Y]$$
for all $x$, that is, $E[Y\mid X]$ is a degenerate random
variable that equals the number $E[Y]$ with probability $1$.

In your particular example, BabakP's answer after correction
by Moderator whuber shows that $E[Y\mid X = x]$ is a random
variable that takes on values $1, 1, 2$ with probabilities
$0.25, 0.25, 0.5$ respectively and so its expectation is
$0.25\times 1 + 0.25\times 1 + 0.5\times 2 = 1.5$ while the
$Y$ itself is a random variable taking on values $1$ and $1$
with equal probability $0.5$ and so $E[Y] = 1\times 0.5 + 2\times 0.5 = 1.5$
as indeed one expects from the law of iterated expectation
$$E\left[[Y\mid X]\right] = E[Y].$$ 
If the joint pmf was intended
to illustrate the difference between conditional
expectation and expectation, then it was a spectacularly bad choice
because the random variable $E[Y\mid X]$ turns out to have the
same distribution as the random variable $Y$, and so the expected
values are necessarily the same. More generally, $E[Y\mid X]$ does
not have the same distribution as $Y$ but their expected values are
the same.
Consider for exampple, the joint pmf
$$\begin{array}{c c c c}
& \quad    &X=1\quad   &X=-1\quad  &X=2  \\
&Y=1\quad  &0.2\quad  &0.2\quad  &0.1   \\
&Y=2\quad  &0.2\quad     &0.1\quad     &0.2
\end{array}$$
for which the conditional pmfs of $Y$ are
$$X=1: \qquad p_{Y\mid X}(1\mid X = 1) = \frac{1}{2}, p_{Y\mid X}(2\mid X = 1) = \frac{1}{2}\\
X=-1: \qquad p_{Y\mid X}(1\mid X = 1) = \frac{2}{3}, p_{Y\mid X}(2\mid X = 1) = \frac{1}{3}\\
X=2: \qquad p_{Y\mid X}(1\mid X = 1) = \frac{1}{3}, p_{Y\mid X}(2\mid X = 1) = \frac{2}{3}$$
the conditional means are
$$\begin{align}
E[Y\mid X = 1] &= 1\times \frac{1}{2} + 2 \times \frac{1}{2} = \frac{3}{2}\\
E[Y\mid X = -1] &= 1\times \frac{2}{3} + 2 \times \frac{1}{3} = \frac{4}{3}\\
E[Y\mid X = 2] &= 1\times \frac{1}{3} + 2 \times \frac{2}{3} = \frac{5}{3}
\end{align}$$
that is, $E[Y\mid X]$ is a random variable that takes on values
$\frac{3}{2}, \frac{4}{3}, \frac{5}{3}$ with probabiliities
$\frac{4}{10}, \frac{3}{10}, \frac{3}{10}$ respectively which is
not the same as the distribution of $Y$. Note also that $E[Y] = \frac{3}{2}$
happens to equal $E[Y\mid X=1]$ but not the
other two conditional expectations.  While $E[Y\mid X]$ and $Y$ have
different distributions, their expected values are the same:
$$E\left[E[Y\mid X]\right] = \frac{3}{2}\times\frac{4}{10}
+\frac{4}{3}\times\frac{3}{10} + \frac{5}{3}\times \frac{3}{10}
= \frac{3}{2} = E[Y].$$
</s>