How to show $E[X\mid X = x] =x$?

Question

I read that $E[X\mid X = x] =x$ but I don't get that when I try to prove it: \begin{align} E[X\mid X = x] &= \sum x P(X=x|X=x) \\ &= \sum x \frac{P(X=x,X=x)}{P(X=x)} \\ &= \sum x \frac{P(X=x)}{P(X=x)} \\ &= \sum x \\ \end{align} So what am I doing wrong?

Answer 1

You are using $x$ for two different purposes: as a free variable and as a bound variable (index). The more correct solution (at least for a discrete random variable), using $x$ as a free variable and another variable ($y$) as a bound variable:

$$ \begin{align} E[X|X = x] &= \sum_y y P(X=y|X=x) \\ &= \sum_{y\ne x} y \frac{P(X=y,X=x)}{P(X=x)} + x \frac{P(X=x,X=x)}{P(X=x)}\text{ (split out }y=x\text{ from }y\ne x\text{)} \\ &= x\frac{P(X=x)}{P(X=x)}\text{ (all terms in the sum above are zero)} \\ &= x \\ \end{align} $$

Answer 2

You indexed wrong:

$$\mathbb E(X|X=x) = \sum_k k \mathbb P(X=k|X=x) = x \mathbb P(X=x|X=x) = x $$

Answer 3

$$\mathbb{E}[X|X=x]=\mathbb{E}[x]=x$$

When you say $X=x$ it means that $x$ is a fixed value, not anymore a rv