Let X 1 , X 2 be a random variables with geometric distribution with p,Find...

Baladdaa9

Baladdaa9

Answered

2022-07-20

Let X 1 , X 2 be a random variables with geometric distribution with p,
Find the probability function of the random variable X 1 + X 2

Answer & Explanation

slapadabassyc

slapadabassyc

Expert

2022-07-21Added 21 answers

Step 1
The post contains some elements of a correct approach.
We cannot solve the problem without making some assumptions about the relationship between X 1 and X 2 . We will assume that X 1 and X 2 are independent.
Let S = X 1 + X 2 . We want to find Pr ( S = n ). This is 0 if n 1. So let n 2.
Step 2
We can have S = n in several ways. For we could have X 1 = 1 and X 2 = n 1. Or we could have X 1 = 2 and X 2 = n 2. And so on up to X 1 = n 1 and X 2 = 1.
For any k from 1 to n 1, we have Pr ( X 1 = k X 2 = n k ) = ( 1 p ) k 1 p ( 1 p ) n k 1 p = p 2 ( 1 p ) n 2 ..
Adding up from k = 1 to k = n 1 we find that Pr ( S = n ) = ( n 1 ) p 2 ( 1 p ) n 2 .

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