Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 9 / a

A probability density function must be nonnegative and integrate to one. For $c>0$,At $c=0$ the function is identically zero, while $c<0$ makes it negative on $x>1$. Thus $\boxed{c>0}$ is necessary and sufficient. The resulting law is a Pareto distribution with scale one and shape $c$.

Integrating the probability density function from the lower endpoint gives the cumulative distribution functionThe cumulative distribution function is continuous at $x=1$ and tends to one at infinity.

For $c>1$, the expected value is the convergent improper integralFor completeness, when $0<c\le 1$ the same positive integral diverges and the expected value is infinite.

The logarithm is strictly increasing, so $\mathbb{P}(Y\le y)=\mathbb{P}(X\le e^y)$. Since $X>1$ almost surely,Thus the logarithm of a Pareto random variable has an exponential distribution with rate $c$. Its probability density function value at the single endpoint can be chosen arbitrarily without changing the law.