Questions and Answers of Principles Of Uncertainty

1. We know from Theorem 5.4.10 that if an n × n matrix A satisfies | A |= 0, then there is some vector x, x = 0 such that Ax = 0. We also know from Corollary 5.7 2 that if matrix A has a row of
6. Prove or disprove: The set of permutations β of {1, 2, . . . , n} such that sgn(β) = −1 form a subgroup.
5. Determine whether the following sets and operations form a group:(a) the positive integers under addition(b) all integers (positive, negative and zero) under addition(c) all integers under
4. Using the same setup as problem 3(a) compute sgn (β1β2) directly(b) compute sgn (β2β1) directly(c) show sgn (β1β2) = sgn (β2β1)
3. Let n = 3, and let β1(1, 2, 3) = (1, 3, 2) and β2(1, 2, 3) = (2, 1, 3).(a) Compute β1β2.(b) Compute β2β1.(c) Show that β1β2 = β2β1.
2. For n = 3,(a) What are the six possible permutations?(b) For each of them, apply (5.20) to find its signature.
1. Vocabulary. State in your own words the meaning of:(a) permutation(b) signature of a number(c) signature of a permutation
2. Prove the following about inner products:(a) < x, y >=< y, x >(b) < ax, y >= a < x, y > for any number a(c) < x, y + z >=< x, y > + < x, z >
1. Vocabulary. Explain in your own words:(a) linear space(b) span(c) linear independence(d) basis(e) finite dimensional linear space(f) inner product, length, distance(g) orthogonal vectors(h)
2. Let X1 and X2 be independent random variables each having the distribution?
1. Let X have a Poisson distribution with parameter λ. Suppose Y = X2. Find the distribution of Y . Is g one-to-one?
2. Consider the sequence of independent random variables defined by Xn =�n with probability 1/n . 0 with probability 1 − 1/n(a) Does Xn converge almost surely? If so, to what random variable does
1. Vocabulary. Explain in your own words:(a) σ-field(b) convergence in probability(c) almost sure convergence(d) weak law of large numbers(e) strong law of large numbers
6. Prove (4.111).
5. (a) Prove (4.96).(b) Use (4.96) to prove partb) of Lemma 4.9.6.
4. (a) Prove (4.90).(b) Use (4.90) to show that, if f ≤ g, (4.85) holds.
3. (a) Prove (4.89).(b) Use (4.89) to show that cf EM(A, α) and that (4.84) holds.
2. Why is Cousin’s Lemma important? If it were not true, what consequences would that have?
1. Vocabulary. Explain in your own words:(a) partition(b) δ-fine partition(c) cell(d) McShane-Stieltjes (or McShane) integral(e) Cousin’s Lemma
2. Consider the following distribution for the uncertain quantity P, that indicates my probability that a flipped coin will come up heads. With probability 2/3, I believe that the coin is fair, (P =
1. (a) Vocabulary. Explain in your own words what the Riemann-Stieltjes integral is.(b) Why is it useful to think about?
3. Example 1 after Corollary 4.7.9 displays a sequence of functions fn(x) that converge to a limiting function f(x).(a) Use the definition of uniform convergence to examine whether this convergence
2. In section 3.9 the following example is given:Let Xn take the value n with probability 1/n, and otherwise take the value 0. Then E(Xn) = 1 for all n. However limn→∞ P{Xn = 0} = 1, so the
1. Vocabulary. Explain in your own words:(a) Riemann probability(b) Riemann expectation(c) Weak countable additivity(d) Strong countable additivity
2. In Example 1, what is fn(x)dx? Show that it is a Cauchy sequence. What is its limiting value?
1. Vocabulary. Explain in your own words:(a) Riemann integrability(b) bounded convergence(c) dominated convergence
2. Use the definition of Riemann integral to find � xdx. Hint: You may find it helpful to 0 review section 1.2.2.
1. Vocabulary: state in your own words what is meant by:(a) Riemann sum(b) Riemann integral(c) Step function
5. Consider the following two statements about a space X :(a) For every xEX , there exists a yEX such that y = x.(b) There exists a yEX such that for every xEX , y = x.i. For each statement, find a
4. Let U ≥ L. Let x1, x2, . . . be a sequence that is convergent but not absolutely convergent.Show that there is a reordering of the x’s such that U is the limit superior of the partial sums of
3. Prove the following: Suppose bn is a non-increasing bounded sequence. Then bn has a limit.
2. Consider the three examples given just after the proof of Theorem 4.7.2. For each of them, identify the limit superior and the limit inferior.
1. Vocabulary. Explain in your own words:(a) accumulation point of a set(b) accumulation point of a sequence(c) Cauchy criterion(d) limit superior(e) limit inferior
2. Reconsider problem 3 of section 4.2, continued in problem 3 of section 4.3.(a) Find the conditional expectation and the conditional variance of Y given X.(b) Find the covariance of X and Y .(c)
1. Reconsider problem 2 of section 4.2, continued in problem 2 of section 4.3.(a) Find the conditional expectation and the conditional variance of Y given X.(b) Find the covariance of X and Y .(c)
3. Reconsider problem 3 of section 4.2.(a) Find the conditional probability density of X given Y , fX|Y (x | y).(b) Find the cumulative probability density of X given Y , FX|Y (x | y).(c) Use your
2. Reconsider problem 2 of section 4.2.(a) Find the conditional probability density of Y given X: fY X(y | x). |(b) Find the cumulative conditional probability density of Y given X: FY |X(y | x).(c)
1. Vocabulary: State in your own words, the meaning of:(a) the conditional density of Y given X.(b) independence of continuous random variables.
3. Suppose X and Y are continuous random variables having the probability density�k | x + y | −1 < x < 1, −2 < y < 1 fX,Y (x, y) = 0 otherwise(a) Find k.(b) Find the marginal probability
2. Suppose X and Y are continuous random variables that are uniform within the unit circle, that is�c if x2 + y2 1 fX,Y (x, y) = ≤0 otherwise(a) Find c.(b) Find the marginal probability density
1. Vocabulary. Define in your own words:(a) Riemann probability(b) joint cumulative distribution function(c) marginal cumulative distribution function(d) joint probability density function(e)
3. For each of the functions f(x) in problem 2 that satisfies the conditions to be a probability density, find the cumulative density function.
2. Show wheth⎧er each of the following satisfy the conditions to be a probability density:⎪⎨0 x < 0(a) f(x) = ⎪x 0 ≤ x ≤ 2⎩�0 x > 2 1/2 −1 < x < 1 (b) f(x) =�0 otherwise 2x 1 < x
1. Vocabulary. Explain in your own words:(a) probability density function(b) absolutely continuous random variable
2. Use the cdf of X found in exercise 1 to find the expectation of X. Check your answer against the expectation of X where X has a binomial distribution found in exercise 8 of section 2.1.2.
1. Suppose X has a binomial distribution with n = 2, for some p, 0 < p < 1. Find the cdf of X.
3. Recall the rules for “Pick Three” from the Pennsylvania Lottery (see problem 3 in section 1.5.2). Suppose that 2000 players choose their three digit numbers independently of the others on a
2. Suppose that a certain disease strikes .01% of the population in a year, and suppose that occurrences of it are believed to be independent from person to person. Find the probability of three or
1. Suppose that X1 and X2 are independent Poisson random variables with parameters λ1 and λ2, respectively.(a) Find the probability generating function of X1 + X2.(b) What is the distribution of X1
5. Do the same problem as 4 when there are a apples, b bananas and c cantaloupes in the bowl. Find the distribution of the number of fruit selected before the a∗th apple is selected, for each a∗,
4. Hypergeometric Waiting Time. Suppose a bowl of fruit contains five apples, three bananas and four cantaloupes. Suppose these fruits are sampled without replacement, choosing each fruit equally
3. Suppose X has a negative binomial distribution with parameters r and p, and Y has a negative binomial distribution with parameters s and p. Show that X +Y has a negative binomial distribution with
2. Prove your answer to problem 1.
1. Is it reasonable to suppose that the negative binomial distribution has the memoryless property? Why or why not?
4. (Rudin (1976, p. 196)) Consider the following two-dimensional array of numbers
3. Let T ∗ = ∞ 1 i=1 i3 .(a) Show that T ∗ < ∞.(b) Define W as follows:1 P{W = i} = i = 1, 2, . . . T ∗i3 Show that W is a random variable.(c) Show that E(W) exists.(d) Show that E(W2) does
2. Let T = ∞ 1/i2 i=1 . Recall 0 < T ≤ 2.Let Y be defined as follows:⎧⎪⎨Y = i, if i is even, with probability 1/Ti 2y = = −i, if i is odd, with probability 1/Ti 2= 0 otherwise(a) Show
1. Vocabulary: Explain in your own words:(a) convergent series(b) absolutely convergent series
4. Verify p1,+ = 1/2 from the example.
3. Calculate P{X = 1, Y = 8} and P{X = 0, Y = 8} in the example discussed in this section.
2. Make your own example in which the conglomerative property fails.
1. Vocabulary: Explain in your own words:(a) dynamic sure loss(b) conglomerability(c) cofinite set(d) residue class, mod k(e) partition(f) uniform distribution Why are these important?
4. Show that the set of real numbers satisfying a < x < b for some a andb, can be put in one-to-one correspondence with the set of real numbers y satisfying c < y < d for every c and d.
3. Show that the set of positive real numbers can be put in one-to-one correspondence with the set of real numbers x, 0 < x < 1. Hint: think about the function g(x) = 1 − 1. x
2. Find mappings that show that each of the following is countable:(a) the positive and negative natural numbers . . . − 3, −2, −1, 0, 1, 2, 3, . . .(b) all rational numbers, both positive and
1. Vocabulary: Explain in your own words the meaning of:(a) natural number(b) rational number(c) real number(d) countable set
1. Vocabulary. Explain in your own words the following:(a) event(b) sure event(c) disjoint events(d) exhaustive events(e) the union of two events(f) sure loser(g) coherent(h) probability
2. Consider the events A1, A2, A3 and A4 defined in the beginning of section 1.1.1 and as applied to your current geographic area for tomorrow. What prices would you give for the tickets? Explain
3. (a) Suppose that someone offers to buy or sell tickets on the events A1 at price $0.30, on A2 at price $0.20, and on the event of rain at price $0.60. What purchases and sales would you make to
4. Think of something you are uncertain about. Define the events that matter to you about it. Are the events you define disjoint? Are they exhaustive? Give your prices for tickets on each of those
5. Suppose that someone offers to buy or sell tickets at the following prices:If the home team wins the soccer (football, outside the U.S. and Canada) match:$0.75 If the away team wins: $0.20
What purchases and sales would you make to ensure a sure gain for yourself? Show that a sure gain results from your choices. How much can you be sure to gain if you buy or sell no more than four
1. Vocabulary. Explain in your own words:(a) complement of an event(b) empty event(c) several disjoint events(d) the union of several events(e) mathematical induction
2. Consider two flips of a coin, and suppose that the following outcomes are equally likely to you: H1H2, H1T2, T1H2 and T1T2, where Hi indicates Heads on flip i and similarly for Ti.(a) Compute your
3. Consider a single roll of a die, and suppose that you believe that each of the six sides has the same probability of coming up.(a) Find your probability that the roll results in a 3 or higher.(b)
4. (a) Find the sum of the first k even positive integers, as a direct consequence of the formula for the sum of the first k positive integers.(b) From the sum of the first 2k integers, find by
1. Vocabulary. State in your words the meaning of:(a) the intersection of two events(b) Venn Diagram(c) subset(d) element(e) DeMorgan’s Theorem(f) Boole’s Inequality
2. Show that if A and B are any events, AB = BA.
3. Show that, if A, B and C are any events, A(BC) = (AB)C.
4. Show that, if A, B and C are any events, that A(B ∪ C) = AB ∪ AC.
5. Reconsider the situation of problem 2 in section 1.2.6: Two flips of a coin, and the following outcomes are equally likely to you: H1H2, H1T2, T1H2 and T1T2 where Hi indicates heads on flip i and
6. Consider again the weather example of section 1.1, in which there are four events:A1: Rain and High above 68 degrees F tomorrow.A2: Rain and High at or below 68 degrees F tomorrow.A3: No Rain and
1. Vocabulary. Explain in your own words what the limit of a sequence of numbers is.
2. Do all sequences of numbers have a limit? Prove your answer.
3. Let a 2 n = 1/n . Prove limn→∞ an = 0.
4. Let an = 0 if n is odd, and an = 1 if n is even. Does an have a limit? Prove your answer.
5. Let an = (n + 1)/n. Does an have a limit? If so, what is it? Prove your answer.
1. Vocabulary. Explain in you own words what is meant by the conditional probability of A given B.
2. Write out the argument for the case xy < z in the proof of Theorem 2.1.1.
3. Make your own example to show that P{A|B} and P{B|A} need not be the same.
4. Let B be an event such that P{B} > 0. Show that P{· |B} satisfies (1.1), (1.2) and(1.3), which means to show that(i) P{A|B} ≥ 0 for all events A.(ii) P{S|B} = P{B|B} = 1.(iii) Let AB and CB be
5. Suppose that my probability of having a fever is .01 on any given day, and my probability of having both a cold and a fever on any given day is .001. Given that I have a fever, what is my
1. Extend the approximation by calculating the next-order term in the Taylor expansion.Compare the resulting approximation to the approximation discussed above. Is the new approximation more accurate?
2. Compare the approximate and exact solutions to the birthday problem for Martians, both computationally and graphically (see section 2.2.1, exercise 1).
3. Try it for Jovians. Can your computer handle vectors of the lengths required?
4. Create a setting and give numbers to probabilities that lead to Simpson’s Paradox.
5. In your judgment, do the data in Table 2.3 indicate underrepresentation of Maoris on New Zealand juries? Does your answer depend on whether New Zealand juries are chosen to represent the entire
6. In section 2.3, Simpson’s paradox is introduced in terms of an unmeasured variable (in the example, the expertise of the climber). What is the equivalent variable in Table 2.3?How is it possible
1. What are the differences among the three forms of Bayes Theorem?
2. Suppose A1, A2, A3 and A4 are four mutually exclusive and exhaustive events. Also suppose P{A1} = 0.1 P{A2} = 0.2 P{A3} = 0.3 P{A4} = 0.4.Let B be an event such that P{A1B} = 0.05 P{A2B} = 0.15

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