This problem considers the Poisson distribution, a probability distribution for a discrete random variable which was first used by Simeon-Denis Poisson to describe seemingly random criminal events in 1837 in Paris. If independent events have a constant tendency to occur and if the average rate of occurrence is a, then the probability that n events actually occur is given by

with

(a) By noting that

Show that

thereby verifying that the poisson distribution is normalized.

**I've done this already**
(b) By using

and

, show that

thereby verifying that the average rate of occurrence, or the expectation value

, is equal to a.

**I've done this one as well**
(c) By using similar techniques, find

and show, using

and

that the standard deviation of the poisson distribution is given by

**This is the one I need help with**