2 different probability density functions will give the same E(g(x)) or not

[saintman4]

If we have 2 different probability density functions p(x) and q(x). Then we find the expected values of function g(x) by using these 2 different probability density functions. Do both of them give the same expected value?

[HallsofIvy]

If p(x) and g(x) have different means, then obviously they give different expected values for the function g(x)= x! It is certainly possible that two different density functions will give the same expected value for some functions but certainly not for all.