Sunday, September 1, 2019

probability - How do I show that E(g(X))=intiinftynftyg(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))=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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