Let $X = \{ 0,1,2,3 , \ldots , n , \ldots \}$

Essay

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