Expectation of $\mathbb{E}\left[\min\{\min(\xi,a),B\}\right]$

Question

I am having trouble calculating the expectation of the following formula: $$\mathbb{E}\left[\min\{\min(\xi,a),B\}\right].$$ The only random valuable is $\xi$ with $f(\cdot)$ and $F(\cdot)$ as PDF and CDF, respectively.

Thanks!

Answer 1

Let $Y=\min \{\min \{\xi,a\},B\}$. Then $Y >x$ iff $\xi >x$ and $a>x,B>x$. Hence $P(Y>x)=1-F(x)$ if $a>x,B>x$, $0$ otherwise. In other words, $P(Y \leq x)=F(x)$ if $a>x,B>x$ and $1$ otherwise. Note that the distribution of $Y$ has a jump of magnitude $1-F(c)$ at the point $c$. Hence $EY=\int_{-\infty} ^{c} tf(t)dt+c(1-F(c))$ where $c$ is the minimum of $a$ and $B$.