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Is it right to condition a RV on dependent RV?

  1. Oct 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Let there be two discrete random variables:

    [itex]
    X \in \lbrace 1,2,3,4,5,6,7,8,9,10 \rbrace \quad \text{where } P[X] \text{ is uniformly distributed over the sample space of } X \text{.}
    [/itex]

    [itex]
    B = \left\lbrace
    \begin{array}{cl}
    1 & \text{if} \quad X>4 \\
    0 & \text{otherwise}\\
    \end{array}\right.
    [/itex]

    [itex]
    P[B \mid X]=\left\lbrace
    \begin{array}{cl}
    0 & \text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\
    1 & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\
    \end{array}\right.
    [/itex]

    [itex]
    P[X] = {1\over10}
    [/itex]

    [itex]
    P[X,B] = P[B \mid X]P[X] = \left\lbrace
    \begin{array}{cl}
    0 &\text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\
    {1\over 10} & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\
    \end{array}\right.
    [/itex]
    The above should be agreeable. But what about:

    [itex]
    P[X,B] = P[X \mid B]P
    [/itex]

    Since B is dependent on X, is it meaningful or even correct to write an expression for P[X|B]?

    2. Relevant equations

    3. The attempt at a solution
    I think no because the conditional probability will then be recursive.
     
  2. jcsd
  3. Oct 24, 2012 #2

    haruspex

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    It's perfectly ok to discuss the probability of an underlying event given an observation concerning it.
    P[B=0] = 0.4
    P[X=1|B=0] = 1/4
    P[X=1 & B=0] = 0.1
    etc.
     
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