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Thread: Show the following. (Geometric)

  1. #1
    Junior Member universalsandbox's Avatar
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    Show the following. (Geometric)

    Consider that if X and Y are identically independent and geometrically(p) distributed, show the following:

    P( X ≥ Y ) = 1 / ( 2 - p )
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  2. #2
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    Quote Originally Posted by universalsandbox View Post
    Consider that if X and Y are identically independent and geometrically(p) distributed, show the following:

    P( X ≥ Y ) = 1 / ( 2 - p )
    $\displaystyle \Pr(X \geq Y) + \Pr(X \leq Y) - \Pr(X = Y) = 1$.

    By symmetry, $\displaystyle \Pr(X \geq Y) = \Pr(X \leq Y)$.


    Therefore $\displaystyle 2 \Pr(X \geq Y) = 1 + \Pr(X = Y)$.


    $\displaystyle \Pr(X = Y) = \Pr(X = 0, Y = 0) + \Pr(X = 1, Y = 1) + \Pr(X = 2, Y = 2) + \, ....$

    $\displaystyle = p^2 + (1 - p)^2 p^2 + (1 - p)^4 p^2 + \, .... $

    (since X and Y are independent)

    $\displaystyle = p^2 (1 + (1 - p)^2 + (1 - p)^4 + \, .... ) = p^2 \cdot \frac{1}{1 - (1 - p)^2} = \frac{p}{2 - p}$.


    Therefore $\displaystyle 2 \Pr(X \geq Y) = 1 + \frac{p}{2 - p}$

    etc.
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