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))=∫∞−∞g(x)f(x)dx
using the fact that E(X)=∫∞0P(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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