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Conditional distribution for random variable on interval

  1. Feb 4, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the conditional distribution function and density for the random variable X defined on R given that X is in some interval I = (a,b) where P(X in I) > 0. Assume that the density and distribution for the random variable X is known

    2. Relevant equations

    fX|X[itex]\in[/itex]I = P(X[itex]\leq[/itex]x | X[itex]\in[/itex] I) = fX,X[itex]\in[/itex]I(x,x)/fX[itex]\in[/itex]I(x)

    3. The attempt at a solution

    I'm sorry, but my latex skills are very poor so I will try to describe in words what my problem is.
    The problem I'm having is that I know how to calculate the probability of P(X in I) since we just take the integral of the density function over the interval I in question. However, what do I do with the "joint" distribution fX,X[itex]\in[/itex]I(x,x) that I need for the definition of conditional distribution? That is what I can't figure out.

    Thanks in advance!
     
  2. jcsd
  3. Feb 4, 2014 #2

    Ray Vickson

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    The conditional density is the coefficient of ##\Delta x## in the first-order (small-##\Delta x##) expansion of
    [tex]\text{P} \{ x < X < x + \Delta x | a \leq X \leq b \}
    = \frac{\text{P} \{ x < X < x+ \Delta x \: \& \: a \leq X \leq b \}}{\text{P} \{ a \leq X \leq b \}} [/tex]
    For ##x \in (a,b)##, can you figure out what is the numerator, in terms of the probability density function ##f(.)##? Can you figure out the denominator?
     
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