Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 9 / b

Assume $a<b$. An increasing affine map sends the two endpoints to zero and one when $ca+d=0$ and $cb+d=1$. Solving givesIndeed, for $0\le t\le 1$, the cumulative distribution function of the transformed random variable iswhich is the cumulative distribution function of the continuous uniform distribution on $[0,1]$.

A decreasing affine map works too, by swapping the endpoints:These are the two possible choices, since the image interval must have endpoints zero and one. The case $a=b$ cannot be transformed into a nondegenerate continuous uniform distribution by deterministic constants.

Let $W=U^2$. On $0\le U\le 1$, squaring preserves order. Therefore for $0\le w\le 1$,Including the tails gives the cumulative distribution functionDifferentiating on the interior of the support gives the probability density functionThe divergence near zero is integrable: ${\int }_0^1(2\sqrt{w}{)}^{-1}dw=1$. There is no point mass at zero, and the assigned value of a probability density function at a single endpoint does not affect the probability distribution.