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Math Help - prove this claim

  1. #1
    Senior Member
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    prove this claim

    Let  B_{1},B_{2},... be disjoint events with  \cup_{n=1}^{\infty}{B_{n}}=\Omega

    Show that if A is another event and  P(A|B_{n})=p for all n then P(A)=p.


    deduce that if X,Y are discrete random variables then :

    X and Y are independent.
    The conditional distribution of X given Y=y is independent of y.
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  2. #2
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    Note that \begin{array}{rcl}   A &  =  & {A \cap \Omega }  \\<br />
   {} &  =  & {A \cap \left( {\bigcup\limits_n {B_n } } \right)}  \\<br />
   {} &  =  & {\left( {\bigcup\limits_n {\left( {A \cap B_n } \right)} } \right)}  \\<br />
 \end{array} .

    From the given: \left( {\forall n} \right)\left[ {P\left( {A \cap B_n } \right) = P\left( {A|B_n } \right)P\left( {B_n } \right) = pP\left( {B_n } \right)} \right].

    P\left( A \right) = \sum\limits_n {P\left( {A|B_n } \right)P\left( {B_n } \right)}  = p\sum\limits_n {P\left( {B_n } \right) = p} because \sum\limits_n {P\left( {B_n } \right) = 1} (they are pairwise disjoint).
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