Here is the cdf of an absolutely continuous random variable \(X\):

\[ F(x) = \begin{cases} 1-\exp\left(-\sqrt{x}\right)&x\geq 0\\ 0&\text{else}. \end{cases} \]

1. What is \(P(4\leq X < 9)\)?
2. What is the pdf of \(X\)?
3. Compute \(E(X^n)\) for any \(n\in\mathbb{N}\).
4. What is the variance of \(X\)?
5. What is the median of \(X\)?

Hint

All of the moments of this distribution are finite, but nevertheless, it doesn’t have a moment-generating function (MGF). So you’ll hit a dead-end if you try using that to compute the moments. Instead, use LOTUS. It’s a subtle calculation, but it actually works out pretty nice.