Conditional probability distribution with geometric random variablesLet X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that X + Y = n.

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Conditional probability distribution with geometric random variablesLet X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that   X  +  Y  =  n.

ahem37 · Accepted Answer

Step 1I will assume that X, Y are the numbers of trials until the first success, where the probability of success on any trial is p. Then   Pr  (  X  =  j  )  =  Pr  (  Y  =  j  )  =  (  1  −  p      )          j      −      1        p.Let A be the event   X  =  i and let B be the event   X  +  Y  =  n. Then by the definition of conditional probability, we have   Pr  (  A      |    B  )  =            Pr      (      A      ∩      B      )              Pr      (      B      )        .The probability of   A  ∩  B is easy to compute. It is   Pr  (  X  =  i  )  Pr  (  Y  =  n  −  i  ). This is   (  1  −  p      )          i      −      1        p  (  1  −  p      )          n      −      i      −      1        p, which simplifies to   (  1  −  p      )          n      −      2            p    2  .Step 2For the probability that   X  +  Y  =  n, we can use the fact that   Pr  (  X  +  Y  =  n  )  =  Pr  (  X  =  1  )  Pr  (  Y  =  n  −  1  )  +  Pr  (  X  =  2  )  Pr  (  Y  =  n  −  2  )  +  ⋯  +  Pr  (  X  =  n  −  1  )  Pr  (  Y  =  1  )  ..This simplifies to   (  n  −  1  )  (  (  1  −  p      )          n      −      2            p    2  .Divide. We get       1          n      −      1      . The conditional probability is discrete uniform. One can save a little time in the calculation of   Pr  (  X  +  Y  =  n  ) by noting that   X  +  Y has negative binomial distribution: it is the number of trials until the second success.

Let X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that X+Y=n.

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