Expectation maximum between a constant and a random variable

Question

Let $X$ be a random variable. For sake of simplicity assume it is uniformly distributed from $[0,1]$. Let $c$ be a constant in the same interval.

How do I express $E[\max(X,c)]$ in such a case?

Answer 1

$$\operatorname{E}[\max(X,c)] = \operatorname{E}[X \mid X > c]\Pr[X > c] + \operatorname{E}[c \mid X \le c]\Pr[X \le c] = \frac{c+1}{2} \cdot (1-c) + c \cdot c.$$ The first equality is applicable for any distribution on $X$. The second applies specifically to the case where $X \sim \operatorname{Uniform}(0,1)$ with $c \in (0,1)$.

Answer 2

Hint: Let $Y=\max(X,c)$. If $X\lt c$ then $Y=c$. Note that $\Pr(Y=c)=c$.

If $X\ge c$ then, by symmetry, $Y$ has expectation $\frac{1+c}{2}$.