Chapter 3.1, Problem 20E

Nicol (see References) lets the pdf of X be defined by $f\left(x\right)=\{\begin{array}{cc}\begin{array}{c}x,\\ \frac{c}{x^3}\\ 0,\end{array}& \begin{array}{c}0\le x\le 1,\\ 1\le x<\infty \\ \text{elsewhere}\text{.}\end{array}\end{array}$

Find

(a) The value of c so that $f\left(x\right)$ is a pdf.

(b) The mean of X (if it exists).

(c) The variance of X (if it exists).

(d) $P\left(\frac{1}{2}\le X\le 2\right)$.

(e) ${\pi }_{.82}$.