Question

We roll two six-sided fair dice and let X be the randorn variable corresponding to the maximum value.

1. Compute X(s) for all s in S.

2. Write a formula for PX E B), for any subset B of the real numbers.

Answer 1

1. X → maximum value

Let S is the sample space of too roll.

S ={(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),

(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

=> n(S) = 36

X can take values

X(S) = 1, 2, 3, 4, 5, 6

2. This is PMF P(X $\in$ B)

where B = (a, b)

$P(X \in B) = P(X \in (a, b)) \\ P(a≤X≤b) = P(b) - P(a)$