What is $E[E[X|Y]|X]$? Could it be further simplified?

Question

I am trying to figure out what $E[E[X|Y]|X]$ equals to. Could it be further simplified?

My initial guess is that it should be equal to X. However, I have no clue if it is correct.

I know $E[X|Y] = \int xf(x|y)\,dx$ and $E[E[X|Y]|X] = \int \int xf(x|y)\,dx f(y|x)\,dy$, but I am stuck here.

Answer 1

You are on the right track. To simplify 𝐸[𝐸[𝑋|𝑌]|𝑋], you can use the law of iterated expectations, which states that 𝐸[𝐸[𝑋|𝑌]|𝑋]=𝐸[𝑋|𝑋]. This means that the conditional expectation of 𝑋 given 𝑌, taken over all possible values of 𝑌, is equal to the unconditional expectation of 𝑋, taken over all possible values of 𝑋.

Using this result, we have:

𝐸[𝐸[𝑋|𝑌]|𝑋] = 𝐸[𝑋|𝑋]

Now, we can simplify 𝐸[𝑋|𝑋] using the definition of conditional expectation:

𝐸[𝑋|𝑋] = ∫𝑥𝑓(𝑥|𝑋)𝑑𝑥

where 𝑓(𝑥|𝑋) is the conditional probability density function of 𝑋 given 𝑋. However, 𝑓(𝑥|𝑋) is a Dirac delta function centered at 𝑋, which means that it is equal to 1 when 𝑥=𝑋, and 0 otherwise. Therefore, we have:

𝐸[𝑋|𝑋] = ∫𝑥𝑓(𝑥|𝑋)𝑑𝑥 = ∫𝑥δ(𝑥−𝑋)𝑑𝑥 = 𝑋

where δ(𝑥−𝑋) is the Dirac delta function centered at 𝑋.

Therefore, we have:

𝐸[𝐸[𝑋|𝑌]|𝑋] = 𝐸[𝑋|𝑋] = 𝑋

which confirms your initial guess. So, 𝐸[𝐸[𝑋|𝑌]|𝑋] is equal to 𝑋.