expected value of the maximum of uniform random variable and zero

Question

$X$ is uniformly distributed over $[0,1]$ and $q \in [0,1]$ is a real number. I need to find $\mathbb{E}[\text{max}\{ q-X, 0 \}]$. I would normally split the maximum function into two probabilities, but this would help if I have two uniform distribution. But I have $0$, which I don't know how to consider as a random variable. By the way, I thought that I can simply disregard $0$ and simply look for $\mathbb{E}[ q-X]$, but even then I am not sure how to proceed. I took the probability theory long time ago and got confused.

Answer 1

$\mathbb{E}[\text{max}(q-X,0)] = P(q-X < 0)*0 + P(q-X \geq 0)*\mathbb{E}[q-X|q-X\geq0] $

$ = P(q-X \geq 0) * \mathbb{E}[q-X|q-X\geq0]$

$ = P(X \leq q) * \mathbb{E}[q-X|X\leq q]$

$ = q * \int_{0}^{q} (q-x)\frac{1}{q} dx$

$ = q * \frac{q}{2}$

$ = \frac{q^2}{2}$