 Expected value of a capped nonnegative random variable

For a nonnegative random variable $Y$ and a fixed cap $a\ge 0$, the tail-sum formula givesBelow the cap, the events $\{\min (Y,a)>t\}$ and $\{Y>t\}$ coincide; at and above the cap, the former event is empty. For the second identity, the positive part $(Y-a{)}_{+}$ exceeds $t\ge 0$ precisely when $Y>a+t$. Thus the two integrals divide the area under the survival function, just as $Y=\min (Y,a)+(Y-a{)}_{+}$ divides each outcome. The capped expected value is always finite, while the excess expected value may be infinite.

For the uniform distribution on the interval $[0,4]$ and cap $a=2$, the capped expected value is ${\int }_0^2(1-t/4)dt=3/2$ and the excess expected value is ${\int }_2^4(1-t/4)dt=1/2$. Directly averaging the capped values also gives ${\int }_0^{2}y(1/4)dy+2\mathbb{P}(Y>2)=1/2+1=3/2$. Capping therefore measures a different average from the uncapped expected value two, even when both are finite.