What is the variance of a variable given itself?

Question

Given an event $X$, what is $\operatorname{var}[X\mid X]$. In addition, what would $E[\operatorname{var}(X\mid X)]$ be? I have been told that $\operatorname{var}[X\mid X] = 0$, but I don't understand why - $\operatorname{var}[X\mid X]$ should be a random variable, and $X\mid X$ has no more information than $X$, so $\operatorname{var}[X\mid X]$ should equal $\operatorname{var}[X]$, correct?

Answer 1

By definition, $$ \mathrm{var}[X\mid X=x]=\mathbb{E}[(X-\mathbb{E}[X\mid X=x])^2\mid X=x]$$

But $\mathbb{E}[X\mid X=x]=x$, so $$ \mathbb{E}[(X-\mathbb{E}[X\mid X=x])^2\mid X=x]=\mathbb{E}[(X-x)^2\mid X=x]=0$$

Therefore $\mathrm{var}[X\mid X]=0$.

Answer 2

$X\mid X$ (or more specifically $X\mid X=x$) does have more information than just $X$. If someone tells you that $X$ has a particular, specific value, then how many possible values can $X$ have at that moment? Only one, so the conditional variance must be zero. You can easily verify this algebraically, as carmichael561 does in his answer.

The situation is similar to that of conditional probabilities. The probability that $A$ occurs given that $A$ occurs is unity.