Expectation of the conditional density

Question

What is the difference between E[$X_1$|$X_n$ = $x_n$] and E[$X_1$|$X_n$]? I have found the first one, by integrating x*$f_{X_{(1)}|X_{(n)} = x_{(n)}}$ (x). If anyone has pointers for finding E[$X_1$|$X_n$] I would appreciate it. Thanks!

Answer 1

As you have pointed out, $$\mathbb{E}[X|Y = y] = \int_{-\infty}^{\infty} x f_{X|Y}(x|y)dx.$$ The main difference between $\mathbb{E}[X|Y=y]$ and $Z = \mathbb{E}[X|Y]$ is that $\mathbb{E}[X|Y]$ is a random variable while $\mathbb{E}[X|Y=y]$ is not. Therefore, you can not really 'compute' $\mathbb{E}[X|Y]$ since it is a r.v. Of course, you can compute the expectation of $Z$ by using the tower property, $\mathbb{E}[Z] = \mathbb{E}[\mathbb{E}[X|Y]] = \mathbb{E}[X]$. Similarly, you can compute the variance etc. Also note that if $Y$ is a discrete random variable taking values in the set $\{y_1,\ldots,y_n\}$ then $\mathbb{E}[X|Y = y_j]$ is a realization of $\mathbb{E}[X|Y]$ with probability $\mathbb{P}[Y = y_j]$.