Suppose X and Y are independent Poisson random variables with parameters a and b respectively. Find the joint mass function
P[X = k, Y = m]
(exp(-a).a^k)/k! x (exp(-b).b^m)/m!
Find P[X + Y = n]
I got = P[X = k, Y = n - k]
= sum over k to n (exp(-a).a^k)/k! x (exp(-b).b^n - k)/(n-k)!
But I'm not sure how to complete
is this right so far?
What you have done is not wrong... but it is not very insightful..
Lets start with what you have written..
Lets continue with what you should have done:
Now we can write
Remember Binomial Theorem? It says
This means that
But wait... is actually the distribution of a Poisson R.V with parameter a+b.
Thus if X and Y are independent Poisson random variables with parameters a and b respectively, then X+Y is Poisson with parameter a+b !!
P.S: There is a slick proof of this result using moment generating functions. Try it!
ignore me - I misread what u'd written.. just gonna go over it in detail now.. thanks both! I'll let you know how I get on!
wow.. that's awesome.. thankyou both so much! I have quite a bit of the rest of this question to go.. I'll give it a good go and get back here if there's any probs!