a Give an example of dependent rvs X and Y such that PX

a) Give an example of dependent r.v.s X and Y such that P(X<Y)=1

b) Give an example of independent r.v.s X and Y such that P(X<Y)=1

Solution

a) Take X = number observed from toss of a dice. Let Y = 2X

Clearly P(X < Y) = 1 as both are positive integers

b) For this, we can define X,Y such that maximum value of X is less than minimum of Y.

Simply take X = number observed from toss of a dice.

and take Y = 6 +number observed from a different toss of a dice.

Now X lies in range (1,6) and Y lies in range (7,12)

clearly P(X<Y) = 1, and they both are independent