Chapter 3.1, Problem 8E

For each of the following functions, (i) find the constant c so that $f\left(x\right)$ is a pdf of a random variable X; (ii) find the cdf, $F\left(x\right)=P\left(X\le x\right)$; (iii) sketch graphs of the pdf $f\left(x\right)$ and the distribution function $F\left(x\right)$; and (iv) find $\mu ,{\sigma }^2$, and the index of skewness, $\gamma$:

(a) $f\left(x\right)=\frac{x^3}{4},0<x<c.$

(b) $f\left(x\right)=\left(\frac{3}{16}\right)x^2,-c<x<c.$

(c) $f\left(x\right)=\frac{x}{\sqrt{x}},0<x<1.$ Is this pdf bounded?