1. Poisson Distribution

Let $X$ be a Poisson distribution. Show that $P\{X=k\}$ increases as it approaches to $\lambda$, reaching it's max when k is the largest integer less then $\lambda$ then decrease as $k > \lambda$

Hint: Consider $\frac{P\{X=k\}}{P\{X=k-1\}}$.

since the Poisson distribution is defined as:

$\frac{\lambda^k}{k!}e^{-k}$ I was thinking of using derivatives to max out the equation with respect to k, but realized that due to the factorial I can't since it's not continuous.

I'm not sure I'm doing this right, but if I apply the hint I would get:

$\left( \frac{\lambda^k}{k!}e^{-k} \right) \left( \frac{k!}{\lambda^k}e^{-k} \right) = \left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}$

now I'm not sure what I have to do

2. Originally Posted by lllll
Let $X$ be a Poisson distribution. Show that $P\{X=k\}$ increases as it approaches to $\lambda$, reaching it's max when k is the largest integer less then $\lambda$ then decrease as $k > \lambda$

Hint: Consider $\frac{P\{X=k\}}{P\{X=k-1\}}$.

since the Poisson distribution is defined as:

$\frac{\lambda^k}{k!}e^{-k}$ I was thinking of using derivatives to max out the equation with respect to k, but realized that due to the factorial I can't since it's not continuous.

I'm not sure I'm doing this right, but if I apply the hint I would get:

$\left( \frac{\lambda^k}{k!}e^{-k} \right) \left( \frac{k!}{\lambda^k}e^{-k} \right) = \left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}$

now I'm not sure what I have to do
$\frac{P\{X=k\}}{P\{X=k-1\}}=\left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}$.

Therefore:

$\frac{P\{X=k\}}{P\{X=k-1\}}\ge 1$ when $k\le \lambda$

and:

$\frac{P\{X=k\}}{P\{X=k-1\}} < 1$ when $k > \lambda$

RonL