Chapter 3.1, Problem 7E

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 and $f\left(x\right)$ the cdf $F\left(x\right)$; and (iv) find $\mu ,{\sigma }^2$, and the index of skewness, $\gamma$:

(a) $f\left(x\right)=4x^x,0\le x\le 1.$

(b) $f\left(x\right)=c\sqrt{x},0\le x\le 4.$

(c) $f\left(x\right)=\frac{c}{x^{3/4}},0<x<1.$