1330 words - 6 pages

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 ...view middle of the document...

If not, then the respondent answers randomly(suppose it!). Let Y denote the event people answer “Yes”, and H denote the event the coin lands on the heads side. We know that P (Y ) = 3/5, P (H) = 1/2 and P (Y |H) = 1/2 by assumption. Can you derive P (Y |H c )?

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6. A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, or weak are 0.7, 0.2, and 0.1, respectively. (a) How certain is she will receive the new job oﬀer? (b) Given that she does receive the oﬀer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation? (c) Given that she does not receive the job oﬀer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation? 7. Five percent of the people have high blood pressure. Of the people with high blood pressure, 75 percent drink alcohol; whereas, only 50 percent of people without high blood pressure drink alcohol. What percent of the drinkers have high blood pressure? 8. Let (Ω, F, P) be a probability space and B ∈ F with P (B) > 0. Show that FB = {B ∩ A|A ∈ F} be a σ-ﬁeld on B. Also, show that P (·|B) is a probability measure on (B, FB ). 9. If A and B are independent events, show that the following pairs of events are also independent. (a) A and B c (b) Ac and B (c) Ac and B c 10. Let (Ω, F, P) be a probability space and A, B ∈ F. Suppose that P (A) = 0, P (B) = 0. Prove the following statements. (a) If A and B are independent events, then A ∩ B = ∅. (b) If A ∩ B = ∅, then A and B are not independent. 11. If P (A) = 1/3 and P (B c ) = 1/4, can A and B be disjoint? Explain. 12. Prove each of the following statements.(Assume that any conditioning event has positive probability) (a) If P (B) = 1, then P (A | B) = P (A) for any A. (Caution: B may be a strict subset of Ω.) (b) If A ⊂ B, then P (B | A) = 1 and P (A | B) = P (A)/P (B). (c) If A and B are mutually...

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