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Math Help - Bayes Theorem Limits

  1. #1
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    Bayes Theorem Limits

    Let ( A_n) and ( B_n) be in A with A_n-->A and B_n-->B, with P(B)>0 and P( B_n)>0. Show that:

    a) lim_{n->\infty}P(A_n|B)=P(A|B)

    b) lim_{n->\infty}P(A|B_n)=P(A|B)

    c) lim_{n->\infty}P(A_n|B_n)=P(A|B)

    So, I know that:

    P(A_n|B)= \frac{P(B|A_n)P(A_n)}{P(B)}= \frac{P(B \cap A_n)}{P(B)}

    P(A|B_n)= \frac{P(B_n|A)P(A)}{P(B_n)}= \frac{P(B_n \cap A)}{P(B_n)}

    P(A_n|B_n)= \frac{P(B_n|A_n)P(A_n)}{P(B_n)}= \frac{P(B_n \cap A_n)}{P(B_n)}

    P(A|B)= \frac{P(B|A)P(A)}{P(B)}= \frac{P(B \cap A)}{P(B)}

    My problem really is how to show the limit of P(B|A_n), P(B_n|A), and P(B_n|A_n) is P(B|A). They seem obvious, but I can't figure out how to show it.
    Last edited by azdang; September 16th 2009 at 12:33 PM.
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