Given two iid geometric random variables with p = 1 5 , what is the probability that variates from them equate?

Question

Given two iid geometric random variables with   p  =      1    5  , what is the probability that variates from them equate?

Plutbantonavv · Accepted Answer

Step 1No need for the joint mass function. If the variables count the number of failures before the first success:- The probability that both variates are 0 is             (              1        5            )        2  - The probability that both variates are 1 is             (              1        5            )        2              (              4        5            )        2  - The probability that both variates are 2 is             (              1        5            )        2              (              4        5            )        4  , etc.Step 2Thus the probability that both variates are equal is       ∑          n      =      0        ∞              (              1        5            )        2              (              4        5            )              2      n        =      1    25        ∑          n      =      0        ∞              (              16        25            )        n    =      1    25    ⋅      1          1      −              16        25              =      1    9

Vincent Norman · Accepted Answer

Step 1Think of it this way: You want to find  P  (  X  =  Y  )  =  P  (  X  =  Y  =  1   or   X  =  Y  =  2   or   X  =  Y  =  3   or   .  .  .  )These are clearly disjoint cases, so we break apart into a sum of probabilities while also realizing that each such statement   X  =  Y  =  n is equivalent to saying   X  =  n AND   Y  =  n. Thus we have   P  (  X  =  Y  )  =  P  (  X  =  1  ,  Y  =  1  )  +  P  (  X  =  2  ,  Y  =  2  )  +  P  (  X  =  3  ,  Y  =  3  )  +  .  .  .Step 2Now, we use the fact that the two variates were considered IID. As such, each term   P  (  X  =  n  ,  Y  =  n  )  =      P    2    (  X  =  n  ) and we ultimately end up with the infinite sum   P  (  X  =  Y  )  =      P    2    (  X  =  1  )  +      P    2    (  X  =  2  )  +      P    2    (  X  =  3  )  +  .  .  .

Given two iid geometric random variables with p=1/5, what is the probability that variates from them equate?

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