Suppose that random variables $X\sim\text{Unif}(0,1)$, and assume that the conditional distribution $Y\mid X = x\sim\text{Binom}(n, p = x)$. Compute $E[Y]$.
Given $X, Y\mid X$, find $E[Y]$.
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probability-theory
2 Answers
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Let me observe that your question shows no effort to find the answer by yourself. The best way to do that is to show us the work you have already done and where you are stuck. See here for more.
Conditional on each $x$, your $Y$ has $\mathbb{E}[Y|x]=nx$. Now you simply integrate $nx$ over pdf of $\mathcal{N}[0,1]$.
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The Law of Total Expectation says that $E[Y] = E[E[Y\mid X]]$ (for any r.v. where $|E[X]|<\infty$).
Now, we have that $E[Y\mid X] = np = nX$, and then: $$E[Y] = E[E[Y\mid X]] = E[nX] = nE[X] = \frac{n}{2}$$