Geometric distribution with multiple successes"A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X = the number of doors he knocks on.Find the probability mass function of X"My thought is that x cannot be less than 2, since he would have to knock on two doors to make two sales.I'm thinking the function would be ( x 2 ) ( u 2 ) ( 1 − u ) x − 2 . But when I go to find E(x), that doesn't lend itself well to the geometric form I've learned to love.Am I on the right track at least?

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Geometric distribution with multiple successes&quot;A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X = the number of doors he knocks on.Find the probability mass function of X&quot;My thought is that x cannot be less than 2, since he would have to knock on two doors to make two sales.I&#039;m thinking the function would be                               (                            x        2                              )                      (          u      2        )    (    1    −    u          )              x        −        2              .  But when I go to find E(x), that doesn&#039;t lend itself well to the geometric form I&#039;ve learned to love.Am I on the right track at least?

mainzollbtt · Accepted Answer

Step 1Let us find the probability that   X  =  x, that is, the probability she has to knock on x doors.Step 2She has to have 1 success (and therefore   x  −  2 failures) in the first   x  −  1 trials, and then a success. The probability of this is                     (                            x        −        1            1                      )              u  (  1  −  u      )          x      −      2        u  .

Kayley Dickson · Accepted Answer

Step 1The negative binomial distribution of trials until the second success is:      P    (      X    2    =  x  )  =                              (                                      x          −          1                1                              )                          u    2    (  1  −  u      )          x      −      2      There must be   x  −  2 failures and one success in any order before the last success.Step 2The negative binomial distribution of trials until the r-th success is:      P    (      X    r    =  x  )  =                              (                                      x          −          r          +          1                          r          −          1                                      )                          u          r        (  1  −  u      )          x      −      r      The geometric distribution is the special case of a negative binomial where   r  =  1.      P    (      X    1    =  x  )  =                              (                            x        0                              )                          u          1        (  1  −  u      )          x      −      1        =  u  (  1  −  u      )          x      −      1

A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X= the number of doors he knocks on. Find the probability mass function of X

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