Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 7 / a

Call the normalization constant $K$ to distinguish it from the moment index used later. A probability density function must have total integral one, soTherefore $K=\theta c^{\theta }$, the value denoted $k$ in the density. This is the Pareto distribution with scale $c$ and shape $\theta$.

For $x\le c$, the cumulative distribution function is zero. For $x>c$, integrate the normalized probability density function:It is continuous at $c$, increases to one, and is the cumulative distribution function of the stated Pareto distribution.

Write $r$ for a positive integer moment index. The moment isThis improper integral converges exactly when $r-\theta -1<-1$, namely $r<\theta$. For those indices,At $r=\theta$ the divergence is logarithmic; for $r>\theta$ it is a power divergence. Thus, in the source's moment notation, the positive integers $k$ allowed are exactly $k<\theta$. If $\theta \le 1$, no positive integer moment is finite.