Suppose $X~U(0,1)$ and that conditional on $X$, the random variable $Y~(0,X)$, Find $E[Y]$.

Question

I understand that $E[Y]=E[E(Y|X)]$ but am not sure how to proceed.

Answer 1

You should use the definition of $E[Y|X]=g(X)$, where $g(x)=E[Y|X=x]$

The variable $Y|X=x$ is distributed as $U(0,x)$ so $E[Y|X=x]=\frac{x}{2}$

So $E[Y|X]=\frac{X}{2}$

So $E[Y]=E[E[Y|X]]=E\left[\frac{X}{2}\right]=\frac{1}{2}E[X]=\frac{1}{4}$

Answer 2

You have: $\mathsf E(Y) = \mathsf E(\mathsf E(Y\mid X))$ and also $Y\mid X\sim\mathcal U(0;X)$ and $X\sim\mathcal U(0;1)$

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Well, that $Y\mid X\sim\mathcal U(0;X)$ , should tell you what is $\mathsf E(Y\mid X)$.   (What is the conditional expected value of $Y$ given $X$, when $Y$ is uniform on $(0..X)$ for any given $X$?)

Hint: $\mathsf E(Y\mid X)$ is a random variable that is a function of $X$.

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Then use $X\sim\mathcal U(0;1)$ to determine the expected value of the above conditional expectation.