Conditional probability with geometric distributionX and Y are linearly independent random variables with geometric distribution. They have the same p parameter. How can I calculate the following expressions:a) P ( X = k | X &lt; Y ) = ?b) P ( X = k | X = Y ) = ?c) E ( X ∣ X + Y ) = ?

Question

Conditional probability with geometric distributionX  and Y are linearly independent random variables with geometric distribution. They have the same p parameter. How can I calculate the following expressions:a)   P  (  X  =  k      |    X  &amp;lt;  Y  )  =  ?b)   P  (  X  =  k      |    X  =  Y  )  =  ?c)       E    (  X  ∣  X  +  Y  )  =  ?

Lina Watson · Accepted Answer

Step 1a)                               P                (        X        =        k        ∣        X        &amp;lt;        Y        )                            =                                            P                        (            X            =            k            ,            X            &amp;lt;            Y            )                                              P                        (            X            &amp;lt;            Y            )                                                            =                              (            1            −            p                          )                              k                −                1                                      p                          ∑                              i                =                k                +                1                            ∞                        (            1            −            p                          )                              i                −                1                                      p                                              ∑                              i                =                1                            ∞                                      ∑                              j                =                1                                            i                −                1                                      (            1            −            p                          )                              i                −                1                                      p            (            1            −            p                          )                              j                −                1                                      p                                                            =                              p            (            1            −            p                          )                              2                k                −                1                                                                        1              −              p                                      2              −              p                                                                        =        p        (        2        −        p        )        (        1        −        p                  )                      2            (            k            −            1            )                          .            Step 2b)                               P                (        X        =        k        ∣        X        =        Y        )                            =                                            P                        (            X            =            k            ,            X            =            Y            )                                              P                        (            X            =            Y            )                                                            =                              (            1            −            p                          )                              k                −                1                                      p            (            1            −            p                          )                              k                −                1                                      p                                              ∑                              i                =                1                            ∞                        (            1            −            p                          )              i                        p            (            1            −            p                          )              i                        p                                                            =        p        (        2        −        p        )        (        1        −        p                  )                      2            (            k            −            1            )                          .            Step 3c)                               E                [        X        ∣        X        +        Y        =        n        ]                            =                                            E                                      [              X                                                1                                                  {                  X                  +                  Y                  =                  n                  }                                            ]                                                          P                        (            X            +            Y            =            n            )                                                            =                                            E                                      [              X                                                1                                                  {                  X                  +                  Y                  =                  n                  }                                            ]                                            1            −            2                          (                                                1                  −                  p                                                  2                  −                  p                                            )                                      .

X and Y are linearly independent random variables with geometric distribution. They have the same p parameter. How can I calculate the following expressions: a) P(X=k|X<Y)=? b) P(X=k|X=Y)=? c) E(X∣X+Y) =?

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