Chapter 15.9, Problem 2P

Show that the average value of a random variable $n$ whose probability function is the Poisson distribution ( 9.8) is the number $\mu$ in ( 9.8 ). Also show that the standard deviation of the random variable is $\sqrt{\mu }.$ Hint: Write the infinite series for $e^x$ differentiate it and multiply by $x$ to get $x{e}^x={\displaystyle \sum \left(n{x}^n/n!\right)}$; put $x=\mu .$ To find ${\sigma }^2$ differentiate the $x{e}^x$ series again, etc.