Conditional expectation $\mathbb{E}[Y\mid Z]$ of indicators $Y=\mathbb{I}_{0<x<b}$, $Z=\mathbb{I}_{a<x<1}$

Question

Let $X\thicksim Uni(0,1)$ and $0<a<b<1$.

Our task is to find $\mathbb{E}[Y\mid Z]$ when $Y=\mathbb{I}_{0<x<b}$ and $Z=\mathbb{I}_{a<x<1}$.

I found that $Y,Z$ are dependent variables and $\mathbb{E}[Y\mid Z=z]=\cases{1&z=0\\ \frac{b-a}{1-a}&z=1}$

but I don't understand what should I do from here and even if I'm right.

Answer 1

You are right that

$$ E\left[Y\mid Z=z\right]=\begin{cases}1&,\text{ if }z=0 \\ \frac{b-a}{1-a} &,\text{ if }z=1\end{cases}$$

In other words, for $z\in \{0,1\}$, you have

$$E\left[Y\mid Z=z\right]=\left(\frac{b-1}{1-a}\right)z+1 $$

So in terms of the random variable $Z$, this means

$$E\left[Y\mid Z\right]=\left(\frac{b-1}{1-a}\right)Z+1 \,\,,\,\text{ a.e.}$$