calculating $E(Y)$

#1
suppose \(X\) is random variable non negative with Distribution function \(F_{X}(x)\) and \(Y\) is random variable with Distribution function \(G_{Y}(t)=1-E(e^{-tX}) ,0\leq t\) . how can i calculate \(E(Y)\)
 
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BGM

TS Contributor
#2
First you can easily check that $Y$ is non-negative almost surely by checking its CDF \( G_Y \). Then the expectation of such a random variable can be expressed as

\( E[Y] = \int_0^{\infty} [1 - G_Y(y)]dy \)

Also, the expectation of \( h(X) \) can be expressed as

\( E[h(X)] = \int_0^{\infty} h(x)dF_X(x) \)

With these facts I think you can fill all the details.