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Thread: Discrete Conditional Distribution

  1. #1
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    Discrete Conditional Distribution

    Let X and Y be random variables with joint probability mass function pXY(x,y) =c(1-2^(-x))^y where X=1,2,3,...,N-1 and Y =1,2,3...

    I need to find c.

    All I can think of is the double summation Sum(y= 0 to oo) Sum(0 to N-1) (c(1-2^(-x))^y) but this does not seem doable.

    I have no ideas what I can try next. Can I please get a hint.

    Thanks!!!
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  2. #2
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    Re: Discrete Conditional Distribution

    $f_{XY}(x,y)=(1-2^{-x})^y,~~~x=1,2,\dots,N-1,~~y=0,1,2,\dots$

    $\displaystyle{\sum_{x=1}^{N-1}~\sum_{y=1}^\infty}~f_{XY}(x,y) = \displaystyle{\sum_{x=1}^{N-1}~\sum_{y=1}^\infty}~(1-2^{-x})^y$

    $\displaystyle{\sum_{x=1}^{N-1}}~2^x - 1 = 2^n-n-1$

    so $c = \dfrac{1}{2^N - N - 1}$
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