Chapter 3.5, Problem 66E

Suppose that Y is a random variable with a geometric distribution. Show that

1. a ${\displaystyle {\sum }_{y}p(y)}={\displaystyle {\sum }_{y=1}^{\infty }{q}^{y-1}p}=1$.
2. b $\frac{p(y)}{p(y-1)}=q$, for y = 2, 3, …. This ratio is less than 1, implying that the geometric probabilities are monotonically decreasing as a function of y. If Y has a geometric distribution, what value of Y is the most likely (has the highest probability)?