# calculating $E(Y)$

##### New Member
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
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.