Question: Using the uniform distribution on $[0,1]$, we choose a point \(X\) at random. Then, using the uniform distribution on $[X, 1]$, we choose a point \(Y\) at random. Draw a picture and label \(X\) and $Y .$ It should be clear that \(Y\) is always at least as big as \(X\), i.e., \(Y \geq X\), i.e., there is zero probability that \(Y \[I_{y \geq x}(x, y)= \begin{cases}1 & y \geq x \\ 0 & y The joint probability density function for \(X\) and \(Y\) is

First explain why this formula for \(f(x, y)\) makes sense. Next, show that for this joint pdf,

Hint: do the \(y\) integral first, and break it into two pieces.

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