a Suppose X Y Z are iid uniform on 0 1 Find PX Y Solutiona X

(a) Suppose X, Y, Z are iid uniform on [0, 1]. Find P(X +Y

Solution

a. X+Y<max{X,Y,Z} is possible if and only if Z is maximum. Probability of Z being maximum is 1/3. Therefore the required probability is 1/3.

b. This is possible only if Y is neither maximum nor minimum. Hence the question can be stated as What is the probability that Y is neither maximum nor minimum? Probability of X being maximum or minimum = 2/3. Probability of Z being maximum or minimum = 2/3. Thus the probability that X and Z are maximum and minimum = 2 x 2/3 x 2/3 = 8/9. Therefore, the required probability is 1-8/9 = 1/9.