Suppose U and V are independent and follow the geometric distribution. p ( k ) = p ( 1 − p ) k for k = 0 , 1 , . . .Define the random variable Z = U + V(a) Determine the joint probability mass function P U , Z ( u , z ) = P r ( U = u , Z = z )(b) Determine the conditional probability mass function for U given that Z = n

Question

Suppose U and V are independent and follow the geometric distribution.  p  (  k  )  =  p  (  1  −  p      )    k      for      k  =  0  ,  1  ,  .  .  .Define the random variable   Z  =  U  +  V(a) Determine the joint probability mass function       P                 U      ,      Z        (  u  ,  z  )  =  P  r  (  U  =  u  ,     Z  =  z  )(b) Determine the conditional probability mass function for U given that   Z  =  n

Nadia Berry · Accepted Answer

Step 1                    Pr        (        U        =        u        )        =                                            Pr        (        number of failures before the first success        =        u        )                            =                                            Pr        (                  failure on the first                       u                     trials and success on trial                       #            (            u            +            1            )                          )                            =                                    p        (        1        −        p                  )          u                .                            Pr        (        Z        =        n        )        =                                            Pr                              (                          (        U        =        0                 &amp;amp;                 V        =        n        )         or         (        U        =        1                 &amp;amp;                 V        =        n        −        1        )                                                  or         (        U        =        2                 &amp;amp;                 V        =        n        −        2        )         or         (        U        =        3                 &amp;amp;                 V        =        n        −        3        )                                                  or         (        U        =        4                 &amp;amp;        V        =        n        −        4        )         or         ⋯        ⋯        ⋯                              )                                              =                                    p        ⋅                              (                          p        (        1        −        p                  )          n                                      )                          +                              (                          p        (        1        −        p        )                              )                          ⋅                              (                          p        (        1        −        p                  )                      n            −            1                                                )                          +                              (                          p        (        1        −        p                  )          2                                      )                          ⋅                              (                          p        (        1        −        p                  )                      n            −            2                                                )                                                                            +                              (                          p        (        1        −        p                  )          3                                      )                          ⋅                              (                          p        (        1        −        p                  )                      n            −            3                                                )                          +        ⋯        ⋯                            =                                            (        n        +        1        )                  p          2                (        1        −        p                  )          n                .                                                Step 2                    Pr        (        U        =        u        ∣        Z        =        n        )                            =                              Pr            (            U            =            u                         &amp;amp;                         Z            =            n            )                                Pr            (            Z            =            n            )                          =                              Pr            (            U            =            u                         &amp;amp;                         V            =            n            −            u            )                                Pr            (            Z            =            n            )                                                            =                              p            (            1            −            p                          )              u                        ⋅            p            (            1            −            p                          )                              n                −                u                                                          (            n            +            1            )                          p              2                        (            1            −            p                          )              n                                      =                  1                      n            +            1                          .            Thus all   n  +  1 possible values of U are equally probable.

Suppose U and V are independent and follow the geometric distribution. p(k)=p(1-p)^k for k=0,1,... Define the random variable Z=U+V. (a) Determine the joint probability mass function P_{U,Z} (u,z)=Pr(U=u, Z=z). (b) Determine the conditional probability mass function for U given that Z=n.

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