What is $E[1/X]$ for $f(x) = 3x^2$ when $0<x<1$?

Question

Given a pdf of $f(x) = 3x^2$ for $0 < x < 1$ or $f(x)=0$ otherwise, how to find $E[1/X]$?

I followed the formula:

($E[1/x] = \int f(x) (1/x) dx$

and I got a result of $3/2$ or $1.5$, which is outside of the support. Because it's outside of the support does that mean that my answer is wrong?

Answer 1

not actually, think that the support $(0,1)$ corresponds to the random variable $X$, since your are considering a function $g(X)=1/X$ the expected value will be, as you said, $$\int f_X(x)g(x)dx$$