A question on uniformly distributed random variables over the interval (0, 1).

Question

How to find the approach to solve the following problem:

Let $ (X, Y, Z)$ be independent uniformly distributed random variables over the interval (0, 1). Find $ P(Z≥XY^2)$ . No idea how to start this question. I have knowledge on uniform distribution, but this seems new to me. Need little bit of hint/help. Thanks for the help.

Answer 1

If you note that, given a variable U uniformly distributed in [0,1], and an independent variable V :

1) if V has values in [0,1], then $P(U<V)=E(V)$

2) $E(UV)= \frac{E(V)}{2}$

3) $E(U^2)=\frac{1}{3}$

Then you have $P(Z>XY^2)=1-E(XY^2)=1-\frac{E(Y^2)}{2}=\frac{5}{6}$