Chapter 5.8, Problem 7E

Suppose that W is a continuous random variable with mean $\mu =0$ and a symmetric pdf f(w) and cdf F(w), but for which the variance is not specified (and may not exist). Suppose further that W is such that $P\left(\left|W-0\right|<k\right)=1-\frac{1}{k}$ for $k\ge 1$.

(Note that this equality would be equivalent to the equality in Chebyshev’s inequality if the variance of W were equal to 1.) Then the cdf satisfies

$F\left(w\right)-F\left(-w\right)=1-\frac{1}{w^2},w\ge 1$.

Also, the symmetry assumption implies that $F\left(-w\right)=1-F\left(w\right)$.

(a) Show that the pdf of W is $f\left(w\right)=\{\begin{array}{cc}\frac{1}{{\left|w\right|}^3}& \left|w\right|>1\\ 0& \left|w\right|\le 1\end{array}$.

(b) Find the mean and the variance of W and interpret your results.

(c) Graph the cdf of W.