Homework SourseX1X2X3 are iid random variables each with uniform01 distribu
X1,X2,X3 are i.i.d. random variables each with uniform(0,1) distribution. Find the pdf for the following random variables:
W1 = max(X1,X2,X3),
W2 = min(X1,X2,X3),
W3 = median(X1,X2,X3)
Solution
W1 = max(X1,X2,X3)
so pdf of w1
= P(W1<x)
= P( X1<x and X2<x and X3<x)
= P(X1<x)^3
= x^3
W2 = min(X1,X2,X3)
so, pdf
= P(W2<x)
= P(X1>x and X2>x ad X3>x)
= P(X1>x)^3
= (1-P(X1<x)^3
= (1-x)^3
W3 = median (X1,X2,X3)
median is the middle element, it can be X1 or X2 or X3 with equal probability
hence pdf of W3 = 1/3
