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.
Best Answer
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.