How should I proceed after conditioning on the given info?

[bondking2]

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) = ...??

 

[andrewkirk]

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.

 

[Ray Vickson]

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
$$P(X_1 > x | X_1 < X_2) = \int_{y=-\infty}^{\infty} P(X_1 > x | X1 < y) f_2(y) \, dy.$$
Use the given fact that $X_1, X_2$ are independent (if that is what your abbreviation "idpt" means).