# Thread: Proving a variable has a poission distribution

1. ## Proving a variable has a poission distribution

Hi

I am given that:

P(Y=k)=$\displaystyle \sum$$\displaystyle \binom{n}{k}$p^k.(1-p)^(n-k).e^-x.(x^n/n!)

where the sum is from n=k to infinity

and I need to show that Y is poission distributued with parameter $\displaystyle \lambda$p

I'm not really sure where to start, should I take out all the variables which are not raised to the power of n out of the sum and change $\displaystyle \binom{n}{k}$ to n!/(n-k)!k!?

Any tips to get me started would be much appreciated

2. ## Re: Proving a variable has a poission distribution

Hello,

Yes you can take out the variables where n is not in the expression.

You can also, in a first time, change the summation from n=0 to infinity (since this is the range of a Poisson distribution). It should simplify quite a lot, since in the $\displaystyle n \choose k$, the n! simplifies, the (n-k)! becomes 0!=1 and k! can be pulled out from the sum.

3. ## Re: Proving a variable has a poission distribution

The question is:
P(Y=k)=$\displaystyle \sum$ $\displaystyle \binom{n}{k}$ $\displaystyle p^k$(1-p)^(n-k)$\displaystyle e^{-x}$ $\displaystyle \frac{x^n}{n!}$

where the sum runs from n=k to infinity (sorry i'm rubbish at using latex)

is this right?

P(Y=k)=$\displaystyle \sum$ x^n/(n-k)!k! $\displaystyle p^k$ (1-p)^(n-k) $\displaystyle e^{-x}$

P(Y=k)=$\displaystyle p^k$ $\displaystyle e^{-x}$ $\displaystyle \sum$ x^n/(n-k)!k! . (1-p)^(n-k)

then change the sum so that it runs from n=0 to infinity

P(Y=k)=(p^k)/k! $\displaystyle e^{-x}$ $\displaystyle \sum$
then I'm not sure what I'll have left inside? x^n...

any hints appreciated

4. ## Re: Proving a variable has a poission distribution

After the change of summation, we have $\displaystyle P(Y=k) = \frac{p^k}{k!}e^{-x}\sum_{j=0}^{+\infty}\frac{x^{j+k}}{j!}(1-p)^j = \frac{p^k}{k!}e^{-x}x^k\sum_{j=0}^{+\infty}\frac{((1-p)x)^j}{j!}$.
We recognize the series of $\displaystyle \exp$, and now you an conclude.

5. ## Re: Proving a variable has a poission distribution

Originally Posted by yellowcarrotz

The question is:
P(Y=k)=$\displaystyle \sum$ $\displaystyle \binom{n}{k}$ $\displaystyle p^k$(1-p)^(n-k)$\displaystyle e^{-x}$ $\displaystyle \frac{x^n}{n!}$

where the sum runs from n=k to infinity (sorry i'm rubbish at using latex)

is this right?

P(Y=k)=$\displaystyle \sum$ x^n/(n-k)!k! $\displaystyle p^k$ (1-p)^(n-k) $\displaystyle e^{-x}$

P(Y=k)=$\displaystyle p^k$ $\displaystyle e^{-x}$ $\displaystyle \sum$ x^n/(n-k)!k! . (1-p)^(n-k)

then change the sum so that it runs from n=0 to infinity

P(Y=k)=(p^k)/k! $\displaystyle e^{-x}$ $\displaystyle \sum$
then I'm not sure what I'll have left inside? x^n...

any hints appreciated
Replace n-k by j, that's a way to change the summation index
And if there's just an n, n=j+k

6. ## Re: Proving a variable has a poission distribution

yep i've done it, thank you so much for all your help