Joint Probability of Independent Geometric Random VariablesLet X , Y ∼ G ( p ) be independent Geometric random variables ( p ∈ ( 0 , 1 )). Show that P ( X = Y ) = p / ( 2 − p ).

Question

Joint Probability of Independent Geometric Random VariablesLet   X  ,  Y  ∼  G  (  p  ) be independent Geometric random variables (  p  ∈  (  0  ,  1  )). Show that   P  (  X  =  Y  )  =  p      /    (  2  −  p  ).

Jakob Chavez · Accepted Answer

Step 1Sometimes &quot;geometric distribution&quot; means a distribution supported on {0,1,2,3,…} and sometimes it means a distribution supported on {1,2,3,…}. Assuming the latter, you have                                  Pr        (        X        =        Y        )                            =                                            Pr        (        X        =        Y        =        1        )        +        Pr        (        X        =        Y        =        2        )        +        Pr        (        X        =        Y        =        3        )        +        Pr        (        X        =        Y        =        4        )        +        ⋯                            =                                              p          2                +                  p          2                (        1        −        p                  )          2                +                  p          2                (        1        −        p                  )          4                +                  p          2                (        1        −        p                  )          6                +        ⋯        .            .Step 2Remember that   a  +  a  r  +  a      r    2    +  a      r    3    +  ⋯  =      a          1      −      r        .In this case   a  =      p    2   and   r  =  (  1  −  p      )    2  .So the sum comes to             p              2        −        p            .In the case of the support being {0,1,2,3,…} you&#039;d start the sum with   Pr  (  X  =  Y  =  0  ), but the rest is the same.

Let X,Y∼G(p) be independent Geometric random variables (p in (0,1)). Show that P(X=Y)=p/(2-p).

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