Let $X = \{ 1,2,3 , \ldots , n , \ldots \}$

Essay

Let $X = \{ 1,2,3 , \ldots , n , \ldots \}$ be a discrete random variable with probability density function $f ( n ) = r ( 1 - r ) ^ { n - 1 }$ , where $0 < r < 1$ .(a) Show that $\sum _ { n = 1 } ^ { \infty } f ( n ) = 1$ . Explain the significance of the value 1.(b) The expected value of the random variable X is defined by $E ( X ) = \sum _ { n = 1 } ^ { \infty } n f ( n )$ . Show that $E ( X ) = \frac { 1 } { r }$ . The distribution of X is known as the geometric distribution.