$X$ is a random variable with finite variance. Let $Y=\alpha X + \beta$. Compute $E[(Y-E[Y|X])^2]$.

Question

$X$ is a random variable with finite variance. Let $Y=\alpha X + \beta$. I've been asked to compute $E[(Y-E[Y|X])^2]$. Where do I start with this? Is the iterated conditional expectation formula relevant at all? Is this the expression for the conditional variance, and if so how is the useful to the computation?

Any guidance on what formulas to use would be great (so far I've only been taught about discrete & continuous random variables, expectation, variance, joint and marginal distributions and conditional distributions).

Thanks!

Answer 1

Since $Y$ is $\sigma(X)$-measurable, we have that $E[Y\mid X]=Y$ (provided that $Y$ is integrable). Therefore, $Y-E[Y\mid X]=0$ and hence $E\left[(Y-E[Y\mid X])^2\right]=0$.