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This is a problem from All of Statistics by Wasserman that I have been struggling with for a while.

Problem

Let $X \sim \text{Uniform}(0,1)$. Let $0<a<b<1$. Let \begin{equation} Y = \begin{cases} 1 & 0<x<b\\ 0 \end{cases} \end{equation} \begin{equation} Z = \begin{cases} 1 & a<x<1\\ 0 \end{cases} \end{equation}

a) Are $Y$, $Z$ independent? Why/Why not?

b) Find $E(Y|Z)$. Hint: What values $z$ can $Z$ take?

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With help from the comments above, I believe I figured it out:

Using the fact that $X\sim\text{Uniform}(0,1)$,

$$P(Y=1,Z=1)=P(x\in(0,b)~\land x\in(a,1)) = b-a$$

Similarly,

$$P(Y=1)P(Z=1) = b(1-a) $$

This shows that $Y,Z$ are dependent.

For part (b), following the hint, we have

\begin{align} E(Y|Z=z) &= \sum_{y} y P(Y=y | Z = z) \\ &= \sum_{y} y P(Y=y,Z=z)/P(Z=z)\\ &= 0 + (1)P(Y=1,Z=z)/P(Z=z)\\ &= P(Y=1,Z=z)/P(Z=z)\\ \end{align}

Compute the probabilities $P(Y=1,Z=z)$ and $P(Z=z)$ for $Z=0,1$ to find \begin{align} E(Y|Z=z) = \begin{cases} \frac{b-a}{1-a} & z=1\\ 1 & z=0 \end{cases} \end{align}

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