# Thread: Poisson distribution - help with question!

1. ## Poisson distribution - help with question!

Hello,

I am having problems solving the following question:
Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

I have no idea what to do... Please help

-Thanks

2. Originally Posted by AAZZ
Hello,

I am having problems solving the following question:
Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

I have no idea what to do... Please help

-Thanks
$\displaystyle \displaystyle \sum_{x = 0} ^ \infty x^n \frac{\lambda^x e^{-\lambda}}{x!} = \sum_{x = 0} ^ \infty (x + 1)^n \frac{\lambda^{x + 1} e^{-\lambda}}{(x + 1)!}.$

Fill in the gaps and complete the problem.

3. What do we do with this formula??

4. Originally Posted by stats2010
What do we do with this formula??
The expectation (ha ha, ... expectation) is that you realise that $\displaystyle \displaystyle \sum_{x = 0} ^ \infty (x + 1)^n \frac{\lambda^{x + 1} e^{-\lambda}}{(x + 1)!} = \lambda \sum_{x = 0} ^ \infty (x + 1)^{n-1} \frac{\lambda^{x} e^{-\lambda}}{x!}$ ....

5. Originally Posted by AAZZ
Hello,

I am having problems solving the following question:
Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

I have no idea what to do... Please help

-Thanks
This appears to be part of an assignment that might count towards your final grade. Please see rule #6: http://www.mathhelpforum.com/math-he...ng-151424.html