Chapter 4, Problem 29SE

a.

To determine

Show that $\frac{P\left(X=x\right)}{P\left(X=x-1\right)}=\frac{\lambda }{x}$ if x is a positive integer.

b.

To determine

Show that the most probable value for X is the greatest integer less than or equal to $\lambda$, if X follows Poisson with parameter $\lambda$. That is show that $P\left(X=x\right)\ge P\left(X=x-1\right)$ if and only if  $x\le \lambda$.