Problem 1. In the statements that follow, A and B are events in a sample space S with probability function P, X is a random variable defined on S, and a and b are constants. Mark T (true) if the statement is always true; F (false) if the statement is sometimes false.
1. 0 <= P(A) <= 1.
2. P(The union of A and B) + P(The intersection of A and B) = P(A) and P(B).
3. If A and B are mutually exclusive, then the intersection of A and B = 0.
4. If A and B are independent, then P(The union of A and B) = P(A)P(B)
5. If A = AU ... UA, then P(A) = SUM P(A)
6. P(The union of A and B) = P(A)P(B|A)
7. P(A|B) = P(B|A)
8. E(aX + b) = aE(X) + b
9. Var(aX + b) = aVar(X) + b
10. sigma(x) = a(sigma(x))
This shows how to determine the validity of statements about a given sample space.