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

Question

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

slapadabassyc · Accepted Answer

Step 1The 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 2We 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        .

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

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