DOUBT - please reply ASAP.

[gauri]

question-

if x and y are two independent random variables that follow uniform distribution on [0,1].
let z= min{x,y}

a. find p( z< 0.5 ) = ???

b. find p( z<0.5 | x=0.75 ) = ??

[Amit Goyal]

if X and Y are two independent random variables that follow uniform distribution on [0,1].
let Z= min{X, Y}

Pr(Z < 0.5) = Pr (min{X, Y} < 0.5)
                  = 1 - Pr (min{X, Y} >= 0.5)
                  = 1 - Pr (X >= 0.5, Y >= 0.5)
                  = 1 - Pr (X >= 0.5)Pr (Y >= 0.5)  [Since X and Y are independent]
                  = 1 - 0.25
                  = 0.75

Pr(Z < 0.5|X = 0.75)  = Pr (min{X, Y} < 0.5|X = 0.75)
                                 = Pr (min{0.75, Y} < 0.5|X = 0.75)
                                 = Pr (Y < 0.5|X = 0.75)
                                 = Pr (Y < 0.5)  [Since X and Y are independent]
                                 = 0.5