Sunday, September 1, 2019

probability - How do I show that $ E(g(X))= int_{-infty}^infty g(x)f(x),dx $?

Given that $X$ is a continuous random variable with pdf $f$, and $g(x)$ is a nonnegative function.



How do I show that

$$
E(g(X))= \int_{-\infty}^\infty g(x)f(x)\,dx
$$
using the fact that $E(X) = \int_0^\infty P(X>x)\, dx$.



I attempted to prove this by plugging in g(X) into the second equation instead of just X. And then I took the inverse of g to come up with just a cdf of X, then I rewrote the cdf to its equivalent integral form, giving me an expression with double integral. I have no idea how to move on from here.



Can anyone help me?

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