Manipulation of Variance and quick way to show $\operatorname{Var}(X\mid X=x) = 0 $

Question

It is quite intuitive that $\operatorname{Var}(X\mid X=x) = 0 $ as the variable is no longer "random". It is also fairly simple to do this from the definition of Variance showing that $\mathbb{E}[X^2] = \mathbb{E}[X]^2 = x^2$ But I was wondering how in general we could manipulate conditioned variance and if there is another way to show this :)

Thanks

Answer 1

I assert that: $\mathsf P(X\,{=}\,x\mid X\,{=}\,x)=1$

Therefore: $\mathsf E(X\mid X\,{=}\,x)~{= x\cdot1+0\\=x}$

Then too: $\mathsf {Var}(X\mid X\,{=}\,x) ~{= (x-\mathsf E(X\mid X\,{=}\,x))^2\cdot1+0\\=0}$

Answer 2

For any function, $$E(f(X))=\int_{\mathbb R}f(x)\,d\mu(\{x\})=f(x).$$

So

$$E(X)=x,E(X^2)=x^2,\sigma^2(X)=0.$$