Let $X$be a Poisson random variable with parameter $\lambda$. Show that $PX=i$increases monotonically and then decreases monotonically as$i$increases, reaching its maximum when $i$is the largest integer not exceeding $\lambda$.

Hint: Consider $PX=i/PX=i-1$.

Given in the question that let $X$be a Poisson random variable with parameter $\lambda$.

Consider the fraction $\frac{P(X=i)}{P(X=i-1)}$for some $i\ge 1$We have that

$\frac{P(X-i)}{P(X-i-1)}=\frac{{\displaystyle \frac{{\lambda }^i}{i!}}{e}^{-\lambda }}{{\displaystyle \frac{{\lambda }^{i-1}}{(i-1)!}}{e}^{-\lambda }}=\frac{\lambda }{i}$

for $i\le \lambda$this fraction is greater than or equal to one. so,that means that on this interval we have that PMF is strictly increasing on the other hand, for $i>\lambda$this fraction is strictly smaller than 1. that means that on that interval PMF is strictly increasing,

Hence, we have proved the claimed

$\frac{P(X=i)}{P(X=i-1)}=$$\frac{\lambda }{i}$

For $i\le \lambda$, this fraction is greater or equal to one, so that means that on this interval we have that PMF is strictly increasing. On the other hand, for$i>\lambda$, this fraction is strictly smaller than 1, so that means that on that interval PMF is strictly decreasing..