Thread: Sum of two Poissons (not as easy as it looks!)

I can show that the sum of two Poisson processes of rates a and b is a Poisson process of rate a+b.

Now, find the probability that the arrival of the combined (a+b) Poisson process comes from the process of rate a. I know that the answer must surely be a/(a+b), but how do we actually go about formally proving it?

Originally Posted by Amanda1990

I can show that the sum of two Poisson processes of rates a and b is a Poisson process of rate a+b.

Now, find the probability that the arrival of the combined (a+b) Poisson process comes from the process of rate a. I know that the answer must surely be a/(a+b), but how do we actually go about formally proving it?

I'm not sure what exactly you mean by "the arrival" of a Poisson process, but I'm pretty sure what you need to prove eventually is that if are independent, such that the distribution of is exponential with parameter and the distribution of is exponential with parameter , then . I have seen this done on the forum a few times; for instance you can write, using independence, , etc.