Chapter 6, Problem 6.24P

Consider independent trials, each of which results in outcome $i,i=0,1,\mathrm{...},k$ with probability $p_i,{\displaystyle \underset{i=0}{\overset{k}{\sum }}{}_{p_i=1}}$. Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.

a. Find $P\left\{N=n\right\},n\ge 1$.

b. Find $P\left\{X=j\right\},j=1,\mathrm{...},k$.

c. Show that $P\left\{N=n,X=j\right\}=P\left\{N=n\right\}P\left\{X=j\right\}$.

d. Is it intuitive to you that N is independent of X?

e. Is it intuitive to you that X is independent of N?