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The question is as follows:

Consider the two uniform random variables $X$ on the interval $[0, n]$ and $Y$ on the interval $[0, X]$. What is the expected value of $Y$?

My idea was to find the expected value of $E[X]$ of $X$ and then let $Y$ be on the interval $[0, E[X]]$, but I fear this is horribly wrong. Can somebody please help?

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That's not so wrong at all, actually.   Not quite there, but you do have an intuitive grasp of the problem.


The Law of Iterated Expectation, also known as Law of Total Expectation, or the Tower Property, states:

$$\mathsf E(Y)~=~\mathsf E\big(\mathsf E(Y\mid X)\big)$$

Now you have been told that $Y\mid X ~\sim~\mathcal U[0;X]$ and $X\sim\mathcal U[0;n]$, so you can take it from here.

$$\begin{align}\mathsf E(Y) ~&=~ \mathsf E( X/2) \\[1ex] &= n/4\end{align}$$

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