Chapter 2.1, Problem 3E

For each of the following, determine the constant c so that $f\left(x\right)$ satisfies the conditions of being a pmf for a random variable X, and then depict each pmf as a bar graph:

(a) $f\left(x\right)=\frac{x}{c},x=1,2,3,4.$

(b) $f\left(x\right)=cx,x=1,2,3,\mathrm{...},10.$

(c) $f\left(x\right)=c{\left(\frac{1}{4}\right)}^x,x=1,2,3,\mathrm{....}$

(d) $f\left(x\right)=c{\left(x+1\right)}^2,x=0,1,2,3.$

(e) $f\left(x\right)=\frac{x}{c},x=1,2,3,\mathrm{...},n.$

(f) $f\left(x\right)=\frac{c}{\left(x+1\right)\left(x+2\right)},x=0,1,2,3,\mathrm{...}$

HINT: In part (f), write $f\left(x\right)=c\left[\frac{1}{\frac{\left(x+1\right)-1}{\left(x+2\right)}}\right]$.