Statistics (multivariate Distribution prob)

[Sinistral]

A fair die is cast at random three independent times. Let the random variable Xi be equal to the number of spots that appear on the ith trial i = 1, 2, 3. Let the random variable Y be equal to max(Xi). Find the cdf and the pmf of Y.

[Amit Goyal]

PMF of Y:
f(y) = Pr(Y = y)
f(6) = Pr(Y= 6) = 1 - Pr(Y < 6) = 1 - Pr(X1 < 6, X2 < 6, X3 < 6) = 1 - (5/6)^3
f(5) = Pr(Y= 5) = Pr(Y < 6) - Pr(Y < 5) = (5/6)^3 - (4/6)^3
f(4) = Pr(Y= 4) = Pr(Y < 5) - Pr(Y < 4) = (4/6)^3 - (3/6)^3
f(3) = Pr(Y= 3) = Pr(Y < 4) - Pr(Y < 3) = (3/6)^3 - (2/6)^3
f(2) = Pr(Y= 2) = Pr(Y < 3) - Pr(Y < 2) = (2/6)^3 - (1/6)^3
f(1) = Pr(Y= 1) = Pr(Y < 2) - 0 = (1/6)^3

CDF of Y:
F(y) = Pr(Y ≤ y)
= 0 for y < 1
= (1/6)^3 for 1 ≤ y < 2
= (2/6)^3 for 2 ≤ y < 3
= (3/6)^3 for 3 ≤ y < 4
= (4/6)^3 for 4 ≤ y < 5
= (5/6)^3 for 5 ≤ y < 6
= 1 for y ≥ 6