Suppose X,Y,Z are postive random variable, the pdf are given as f_X(t),
f_Y(t), f_Z(t). and

Z=min(X, Y)

I know the following method to derive the cdf of Z.

P(Z<t)=P(min(X,Y)<t)= 1- ( 1 - P( X<t ) )( 1 - P(Y<t) ).

I am wondering how to evaluate it by using conditonal distribution.

P(Z<t) = P( min(X, Y) < t ) = 1-P( min(X, Y) > t ) = 1 - [ P( X<t; X<Y) +
P( Y<t; Y<X)]

is this correct and how to proceeding?Thanks.
--
ZHANG Yan


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