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We have an absolutely continuous random variable X with $EX=a$, $DX=b^2$ and a probability distribution function F(x).

The question is to find the mathematical expectation and the dispersion of the random variable $Z=-\log(1-F(X))$.

Please, help! Thanks a lot!

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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.

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