# Question about how to merge poisson distribution

gokuls
In general, if A~Po(a) and B~Po(b) are independent random variables, then C = (A+B)~Po(a+b). Can someone please explain the intuition/simple proof of this and a word problem or example would really help to reinforce this concept. Thanks.

Hey gokuls and welcome to the forum.

Hint: Find the MGF of a Poisson distribution. What does it mean to calculate the MGF of A + B if A and B are independent variables? What is the form of the product of the two MGF's? What does this say about the distribution?

gokuls
Hi Chiro,
This concept is used to solve questions as the one following.

The emergency room switchboard has two operators. One operator answers calls for doctors and the other deals with enquiries about patients. The first operator fails to answer 1% of her calls and the second operator fails to answer 3% of his calls. On a typical day, the first and second telephone operators receive 20 and 40 calls respectively during an afternoon session. Using the Poisson distribution find the probability that, between them, the two operators fail to answer two or more calls during an afternoon session.

The two events are independent in the question above and to calculate the probability they both happen, you have to apparently combine the Poisson variables. By the way, what does MGF mean?
Hey gokuls and welcome to the forum.

Hint: Find the MGF of a Poisson distribution. What does it mean to calculate the MGF of A + B if A and B are independent variables? What is the form of the product of the two MGF's? What does this say about the distribution?