Transformation of Random variable $Y=-2\ln(F(x))$

Question

Let $X$ is a continuous Random variable. with strictly increasing function cumulative distribution function $F(x)$. Find and recognise the distribution of random variable $Y=-2\ln(F(x))$. I need some help here please.

Answer 1

It's easy if $F$ is continuous. In that case $P(F(X)\leq t)=P(X\leq F^{-1}(t))=F(F^{-1}(t))=t$ for $0<t<1.$ So, $F(X)\sim U(0,1).$ $P(-2\log F(X)\leq y/2)=P(\log F(X)\geq -y/2)=P(F(X)\geq e^{-y/2})=1-e^{-y/2}$ for $y>0.$ So, $Y\sim Exp(2)$ or $Y\sim \chi^2_2.$

But if $F$ is not continuous, i.e, there is jump in $F$, then $F(X)$ might not be uniformly distributed.