Proving a variable has a poission distribution

Hi :)

I am given that:

P(Y=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 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 to n!/(n-k)!k!?

Any tips to get me started would be much appreciated :)

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 , the n! simplifies, the (n-k)! becomes 0!=1 and k! can be pulled out from the sum.

Re: Proving a variable has a poission distribution

Thanks so much for your reply :)

The question is:

P(Y=k)= (1-p)^(n-k)

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

is this right?

P(Y=k)= x^n/(n-k)!k! (1-p)^(n-k)

P(Y=k)= 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!

then I'm not sure what I'll have left inside? x^n...

any hints appreciated

Re: Proving a variable has a poission distribution

After the change of summation, we have .

We recognize the series of , and now you an conclude.

Re: Proving a variable has a poission distribution

Quote:

Originally Posted by

**yellowcarrotz** Thanks so much for your reply :)

The question is:

P(Y=k)=

(1-p)^(n-k)

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

is this right?

P(Y=k)=

x^n/(n-k)!k!

(1-p)^(n-k)

P(Y=k)=

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!

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

Re: Proving a variable has a poission distribution

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