91Ó°ÊÓ

Given in the question that let $X$be a Poisson random variable with parameter $\mathrm{λ}$.

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

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

for $i≤\mathrm{λ}$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 $i>\mathrm{λ}$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{\mathrm{λ}}{i}$

For $i≤\mathrm{λ}$, 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$i>\mathrm{λ}$, this fraction is strictly smaller than 1, so that means that on that interval PMF is strictly decreasing..