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Math Help - Conditional probability of integer-valued random variables: (solution checking)

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    Senior Member I-Think's Avatar
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    Conditional probability of integer-valued random variables: (solution checking)

    Suppose X and Y are both integer-valued random variables. Let

    p(i|j)=P[X=i|Y=j], q(j|i)=P[Y=j|X=i]

    Show that P(X=i,Y=j) = \frac{p(i|j)}{\sum_{i}\frac{(p(i|j)}{q(j|i)}}

    Solution

    p(i|j)=P[X=i|Y=j]=\frac{P[X=i,Y=j]}{P[Y=j]}
    P[X=i,Y=j]=P[Y=j]p(i|j)

    Consider
    \frac{p(i|j)}{q(j|i)}=\frac{P[X=i]}{P[Y=j]}
    So
    \sum_{i}\frac{p(i|j)}{q(j|i)}=\sum_{i}\frac{P[X=i]}{P[Y=j]}=\frac{1}{P[Y=j]}\sum_{i}P[X=i]}=\frac{1}{P[Y=j]} as \sum_{i}P[X=i]}=1
    So
    P[Y=j]=\frac{1}{\sum_{i}\frac{(p(i|j)}{q(j|i)}}

    So our result is proven
    P(X=i,Y=j) = \frac{p(i|j)}{\sum_{i}\frac{(p(i|j)}{q(j|i)}}
    Last edited by Moo; December 6th 2011 at 12:46 PM. Reason: latex errors
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  2. #2
    Moo
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    Re: Conditional probability of integer-valued random variables: (solution checking)

    Hello,

    I corrected some LaTeX errors and a typo in the way you defined q (it's the conditional probability, not the joint probability as you initially wrote). And your solution is perfect
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