Chapter 5.3, Problem 10E

Let $X_1,X_2,X_3$ denote a random sample of size n= 3 from a distribution with the geometric pmf

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

(a) Compute $P\left(X_1=1,X_2=3,X_3=1\right)$.

(b) Determine $P\left(X_1+X_2+X_3=5\right)$.

(c) If Y equals the maximum of $X_1,X_2,X_3$, find $P\left(Y\le 2\right)=P\left(X_1\le 2\right)P\left(X_2\le 2\right)P\left(X_3\le 2\right)$.