Question

The joint probability density function of two random variables X1 and X2 is defined by f(x1, x2, x3) = 2, 0 < x1 < x2 < 1

Find the conditional distribution of X1 given X2 = x

Answer 1

$f\left( x_1,x_2 \right) =2,0<x_1<x_2<1\\f_{X_1|X_2}\left( y|x \right) =\frac{f\left( y,x \right)}{f_{X_2}\left( x \right)}=\frac{2}{\int_0^x{2dx_1}}=\frac{2}{2x}=\frac{1}{x},0<y<x$