How should I proceed after conditioning on the given info?

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Homework Statement


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.

Homework Equations

The Attempt at a Solution


P(X1 > x1 | X1 < X2) = P(X1 > x1) P(X1 < X2) = ...??
 
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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.
 
bondking2 said:

Homework Statement


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.

Homework Equations

The Attempt at a Solution


P(X1 > x1 | X1 < X2) = P(X1 > x1) P(X1 < X2) = ...??

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