Math Help - 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.

-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.

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

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.