21. For each case below, determine whether or not the real-valued set function P(A) is in fact a probability set function:

a. Sample space S ¼ {1, 2, 3, 4, 5, 6, 7, 8}, Event space U ¼ fA : A Sg , Set Function P(A) ¼

Sx2Aðx=36Þ for A 2 U.

b. Sample space S ¼ [0, 4), Event Space U ¼ fA : A is an interval subset of S, or any set formed by unions, intersections, or complements of these interval subsets}, Set Function P(A) ¼ R x2A exdx for A 2U.

c. Sample Space S ¼ {x : x is a positive integer}, Event Space U ¼ fA : A Sg; Set Function P(A) ¼

Sx2Aðx2 = 105Þ for A 2 U

d. Sample Space S ¼ (0, 1), Event Space U ¼ fA : A is an interval subset of S, or any set formed by unions, intersections, or complements of these interval subsets}, Set Function P(A) ¼ R x2A 12xð1  x Þ

2 dx for A 2 U