Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 9 / b

Translate the inequality inside the cumulative distribution function:Differentiating with the chain rule gives the probability density functionalmost everywhere. Translation shifts the probability distribution without changing its shape or its total probability.

Reversing sign reverses the inequality. Since $X$ has a probability density function, it has no point masses, soDifferentiating the cumulative distribution function givesalmost everywhere: the minus sign outside $F_X$ cancels the minus sign from the chain rule. If point masses were allowed, the cumulative distribution function would instead require the left limit $1-F_X((-y{)}^{-})$.

Since a square is nonnegative, $F_Y(y)=0$ for $y<0$. For $y\ge 0$, the event $X^2\le y$ is $-\sqrt{y}\le X\le \sqrt{y}$. As $X$ has no point masses,Differentiating for $y>0$ gives the contributions of the two inverse branches:The value assigned to the probability density function at $y=0$ is arbitrary. It can be unbounded near $0$, but there is no atom there because $\mathbb{P}(X=0)=0$. Both positive and negative preimages must be included; keeping just one would generally lose probability.

For any real $y$, put $k=\lfloor y\rfloor$. Since $Y$ is integer-valued,Continuity of the cumulative distribution function of $X$ therefore givesThe strict inequality $X<k+1$ matters before using the absence of point masses. The resulting probability distribution is discrete, with probability mass functionThus $Y$ has no ordinary probability density function with respect to length on the real line. Its cumulative distribution function is a step function; its derivative is zero almost everywhere and does not recover its probabilities. The floor function has converted a density into masses on the integers.