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principles of uncertainty

Questions and Answers of Principles Of Uncertainty

3. An Elisa test is a standard test for HIV. Suppose a physician assesses the probability of HIV in a patient who engages in risky behavior (unprotected sex with multiple partners of either sex, or
4. In the following problem, choice “at random” means equally likely among the alternatives.Suppose there are three boxes, A, B and C, each of which contains two coins. Box A has two pennies, Box
5. Phenylketonuria (PKU) is a genetic disorder that affects infants and can lead to mental retardation unless treated. It affects about 1 in 10 thousand newborn infants. Suppose that the test has a
6. Gamma-glutamyl Transpeptidase (GGTP) is a test for liver problems. Among walking, apparently healthy persons, approximately 98.6% have no liver problems, 1% are binge drinkers, 0.2% have a hepatic
1. Vocabulary. Explain in your own words what it means for a set of events to be independent, and to be conditionally independent given a third event.
2. Make your own example to show that pairwise independence of events does not imply independence.
3. Suppose A and B are two independent and disjoint events. Suppose P{B} = 1/2. What is P{A}? Prove your answer.
4. In the example in section 2.5, carefully write out the calculations for P{T2} and P{T2|C1}. Justify each step you make by reference to one of the numbered equations in the book.
5. Suppose you observe two tails in the example just above, but you do not know what coin was used. Apply Bayes Theorem to find the conditional probability that the coin was coin 1.
6. Suppose someone regards 0, 1 and 2 heads as being equally likely in two flips of the same coin and, in the case of exactly one head, considers heads on the first flip to be as probable as heads on
7. Show that S and φ are independent of every other event.
8. (a) Suppose you flip two independent fair coins. If at least one head results, what is the probability of two heads?(b) Suppose the sexes of children in a family are independent, and that boys and
9. Suppose A and B are conditionally independent events given a third event C. Does this imply that A and B are conditionally independent given C? Either prove that it does, or give a counterexample.
10. Suppose A ⊆ B. Find necessary and sufficient conditions on the pair of probabilities(P({A}, P{B}) for A and B to be independent.
12. Suppose that events A and B are that people have diseases a andb, respectively. Suppose that having either disease leads to hospitalization H = A ∪ B. If A and B are believed to be independent
13. (a) If A and B are independent events, prove that A and B are independent.(b) Suppose A1, A2, . . . , An are independent. Prove by induction on n that A ¯ 1, A2, . . . , An are independent.(c)
1. State in your own words what the Monty Hall problem is.
2. Suppose there are three prisoners. It is announced that two will be executed tomorrow, and one set free. But the prisoners do not know who will be executed and who will be set free. Prisoner A
3. Do a simulation in R to study the Monty Hall problem. Run it long enough to satisfy yourself about the probability of success with the “switch” and “don’t switch” strategies.
4. Reconsider the simpler version of the Monty Hall problem, assuming p1 = p2 = p3 = 1/3.Suppose that you have chosen box 1. If the prize is in box 2, Monty Hall must open box
3 and show you that it is empty. Similarly, if the prize is in box 3, Monty Hall must show you that box 2 is empty. But if the prize is in box 1 (so your initial choice is correct), Monty Hall has a
5. Now consider the general case, where it is not necessarily assumed that p1 = p2 = p3 =1/3. If you choose box i and the prize is in box i, Monty Hall has a choice between showing you that box j = i
1. Vocabulary. Explain in your own words:(a) Gambler’s Ruin(b) Geometric Series(c) L’Hˆopital’s Rule
2. When p = 0.45, i = 90 and n = 100, find ai.
3. Suppose there is probability p that A wins a session, q that B wins, and t that a tie results, with no exchange of money, where p + q + t = 1. Find a general expression for ai, and explain the
4. Now suppose that the probability that A wins a session is pi if he has a current fortune of ai, and the probability that B wins is qi = 1 − pi. Again, find a general expression for ai as a
5. Use R to check the accuracy of the approximation in (2.29).
6. Consider the Gambler’s Ruin problem from B’s perspective. B starts with a fortune of n − i, and has probability q of winning a session, and hence p = 1 − q of losing a session.Let bn i −
1. Vocabulary. Explain in your own words:(a) independence of random variables(b) iterated expectations
2. Show that if X and Y are random variables, and X is independent of Y , then Y is independent of X.
3. Show that if A and B are independent events, then IA and IB are independent random variables.
4. Show the converse of problem 3: if IA and IB are independent indicator random variables, then A and B are independent events.
5. Consider random variables X and Y having the following joint distribution:P{X = 1, Y = 1} = 1/8 P{X = 1, Y = 2} = 1/4 P{X = 2, Y = 1} = 3/8 P{X = 2, Y = 2} = 1/4.Are X and Y independent? Prove
6. For the same random variables as in the previous problem, computea) E{X|Y = 1}b) E{Y |X = 2}
7. Suppose P{X = 1, Y = 1} = x, P{X = 1, Y = 2} = y, and P{X = 2, Y = 1} = z,
8. Suppose X1, . . . , Xn are independent random variables. Let m < n, so that X1, . . . , Xm are a subset of X1, . . . , Xn. Show that X1, . . . , Xm are independent.
1. Prove that n jr+ n = n+1 . j,n−j+1,n−(j+1) j+1,n−j
2. Prove the binomial theorem by induction ron rn.
3. Suppose the stronger team in the baseball World Series has probability p = .6 of beating the weaker team, and suppose that the outcome of each game is independent from the rest. What is the
5. Prove the multinomial theorem by induction ron n.r r
6. Prove the multinomial theorem by induction on k.
7. When k = 3, what geometric shape generalizes Pascal’s triangle?
8. Let X have a binomial distribution with parameters n and p. Find E(X).
10. Suppose that the concessionaire at a football stadium finds that during a typical game, 20% of the attendees buy both a hot-dog and a beer, 30% buy only a beer, 20% buy only a hot-dog, and 30%
11. (Continuing problem 10): What is the probability that a random sample of 15 game attendees will buy a hot-dog?(a) Compute this directly.(b) Compute this using Theorem 2.9.1
1. Suppose there is a group of 50 people, from whom a committee of 10 is chosen at random.What is the probability that three specific members of the group, R, S and T, are on the committee?
2. How many ways are there of dividing 18 people into two baseball teams of 9 people each?
3. A deck of cards has four aces and four kings. The cards are shuffled and dealt at random to four players so that each has 13 cards. What is the probability that Joe, who is one of these four
4. Suppose a political discussion group consists of 30 Democrats and 20 Republicans. Suppose a committee of 8 is drawn at random without replacement. What is the probability that it consists of 3
5. Suppose that instead of the Polya Urn Scheme set-up, k1 balls of the color of the ball drawn and k2 balls of the opposite color are added to the urn.(a) Evaluate P{X2 = 1}.(b) Show that P{X1 = 1}
6. Suppose Polya’s Urn consists of balls of several different colors, and k ≥ 0 balls of the color drawn are added each time.Let Xc i = 1 if color c is drawn in the i th draw.(a) Evaluate P{Xc 2
4. Find the variance of a binomial random variable with parameters n and p.
5. Prove Cov(X, X) = V [X].
6. Prove Cov(X, Y ) = Cov(Y, X).
7. Let a and b be integers such that a

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