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I have this question on a homework assignment, and I'm not sure how to solve it.

Assume $X \sim \exp(\lambda=2)$, define $Y=F(x)$, where F(x) is the cdf function of $X$. Calculate the expected value of $Y$, i.e., $E(Y)$.

I thought maybe you could use the Law of the Unconscious Statistician and have the integral of $F(x) * f(x)$? Any help would be appreciated.

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    $\begingroup$ You made a good start. Why not take the next step and evaluate that integral? $\endgroup$ – whuber Jan 21 '16 at 22:12
  • $\begingroup$ I wonder what the derivative of $\frac12 (F(x))^2$ equals? $\endgroup$ – Dilip Sarwate Jan 21 '16 at 22:44
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Yes, you can definitely use LOTUS. Alternatively, you can use the Probability Integral Transform which says that the CDF of a continuous random variable follows the uniform distribution. You might want to see if you can arrive at the same result using both methods.

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