Assignment 1

1. Prove that : P (E ∪ F ∪ G) = P (E) + P (F ) + P (G) − P (E c ∩ F ∩ G) − P (E ∩ F c ∩ G) − P (E ∩ F ∩ Gc ) − 2P (E ∩ F ∩ G). 2. Prove the followings: (a) (∪∞ An )c = ∩∞ Ac . (cf) In order to prove A = B for two sets A and B, you i=1 n n=1 should show that ∀x ∈ A, x ∈ B, and vice versa.) (b) Let F be a σ-field on Ω. If An ∈ F for all n ∈ N, then ∩∞ An ∈ F. n=1 3. An elementary school is offering 3 language classes: one in Chinese, one in Japanese, and one in English. These are open to any of the 100 students in the school. There are 28 students in the Chinese class, 26 in the Japanese class, and 16 in the English class. There are 12 students that are in both Chinese and Japanese, 4 that are in both Chinese and English, and 6 that are in both Japanese and English. In addition, there are 2 students taking all 3 classes. (a) If a student is chosen randomly, what is the probability that he or she is not in any of these classes? (b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class? (c) If 2 students are chosen randomly without replacement, what is the probability that at least one is taking a language class? 4. A CEO wants to decide how much to invest on a project. If the firm invest x, then the probability of success is 1 − e−2x . If the project succeeds, then the firm earns 1, and if not, nothing. Let Y be the earning from the project. The profit function of the firm is given by π(Y |x) = Y − x. (a) Derive the expected profit function, Eπ(Y |x). (b) Suppose that the firm is risk neutral, and therefore wants to maximize the expected profit. How much should they invest? 5. A researcher wants to know how many people have experience in gambling. However, people usually do not answer frankly to such a subtle question. So, she devises the following procedure. First, the respondent tosses a fair coin secretly.