How inequality can be used to derive additional bounds

Question

Question

Let N be a non-negative, integer-valued random variable, Show that

                             P{N > 0} >= (E[N])²/E[N ]²

And explain how this inequality can be used to derive additional bounds on a reliability function.

HINT:                            E[N ]²  = E[N ²|N > 0]P{N > 0}      (Why?)

                                                >= (E[N |N > 0])²P{N > 0}      (Why?)

Now multiply both sides by P{N > 0}.