We have an absolutely continuous random variable $X$ with $\mathbb E[X]=a$, $\mathbb D[X]=b^2$ and a probability distribution function $F(x)$.
The question is to find the mathematical expectation and the dispersion of the random variable $Z=-\log\big(1-F(X)\big)$.
Please, help! Thanks a lot!
Since $F(X)$ is uniformly distributed over $(0, 1)$. Then \begin{align*} E(Z) &= \int_0^1-\ln(1-x)dx\\ &=\int_0^1 \ln x dx =1, \end{align*} and \begin{align*} Var(Z) &= \int_0^1 \ln^2(1-x)dx -1\\ &=\int_0^1\ln^2x dx-1 =1. \end{align*}
Alternatively, note that $Z$ is an exponential random variable with distribution function $1-e^{-x}$, for $x>0$, and $0$, otherwise. We can obtain the same expectation and variance as above.