Returns on investments overseas, especially in Europe and the Pacific Rim, are expected to be higher than those of U.S. markets in the near term, and analysts are now recommending investments in international portfolios. An investment consultant believes that the probability distribution of returns (in percent per year) on one such portfolio is as follows:

x(%)P(x)

9 ..... 0.05

10 ..... 0.15

11 ..... 0.30

12 ..... 0.20

13 ..... 0.15

14 ..... 0.10

15..... 0.05

a. Verify that P(x) is a probability distribution.
b. What is the probability that returns will be at least 12%?

c. Find the cumulative distribution of returns.