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How should I proceed after conditioning on the given info?

  1. Mar 2, 2016 #1
    1. The problem statement, all variables and given/known data
    X1 nad X2 are two idpt r.v. let mu and lamda denote their respective rates. Find the conditional distribution of X1 given X1 < X2.

    2. Relevant equations


    3. The attempt at a solution
    P(X1 > x1 | X1 < X2) = P(X1 > x1) P(X1 < X2) = ....??
     
  2. jcsd
  3. Mar 2, 2016 #2

    andrewkirk

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    Use the usual equation for conditional probabilities
    $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$
    Here you would use
    $$A=\{(X1,X2)|X1>x1\}$$
    $$B=\{(X1,X2)|X1<X2\}$$
    To get the two probabilities on the right-hand side, write down the joint pdf of X1 and X2, then integrate it over suitable regions in the number plane.
     
  4. Mar 3, 2016 #3

    Ray Vickson

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    If ##X_1, X_2## have probability densities ##f_1, f_2##, then
    [tex] P(X_1 > x | X_1 < X_2) = \int_{y=-\infty}^{\infty} P(X_1 > x | X1 < y) f_2(y) \, dy.[/tex]
    Use the given fact that ##X_1, X_2## are independent (if that is what your abbreviation "idpt" means).
     
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