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029Posterior After One Correct and One Incorrect ForecastAn analyst is senior with prior probability 0.3 and junior otherwise. A senior analyst calls a binary market move correctly with probability 0.8 on each day; a junior does so with probability 0.6. On two independent days you observe exactly one correct call and one incorrect call from the same analyst. What is the posterior probability the analyst is senior?概率中等数值题未尝试免费033Three-Regime Bayesian Updating from Daily ReturnsA portfolio manager models the market as being in one of three regimes, each equally likely a priori: - **Bull**: the stock goes up on any given day with probability 4 5 . - **Neutral**: the stock goes up with probability 1 2 . - **Bear**: the stock goes up with probability 1 5 . Days are conditionally independent given the regime. Over three days the stock goes up, up, then down. (a) What is the posterior probability of each regime? (b) What is the conditional probability the stock goes up on Day 4?概率中等数值题未尝试免费040Sequential Signal Updating and the Tower PropertyA quant researcher believes a directional signal has accuracy p that is either 1 3 or 2 3 , each equally likely a priori. On each day the signal independently (given p) predicts the market direction, and is correct with probability p. (a) On Day 1 the signal is correct. What is the posterior P\! (p = \tfrac 2 3 \mid C 1 )? (b) On Day 2 the signal is wrong. Starting from the Day-1 posterior, compute the updated P\! (p = \tfrac 2 3 \mid C 1, W 2 ). (c) Verify the tower property of conditional expectation: show that E[p] = E\! [\,E[p \mid D 1]\, ], where D 1 \in \ C 1, W 1\ denotes the Day-1 outcome. Compute all quantities explicitly.概率困难derivation未尝试面试订阅043Factory Defect Tracing and Predictive InferenceA factory has three production lines with the following output shares and defect rates: | Line | Share of output | Defect rate | |------|----------------|-------------| | 1 | 50% | 2% | | 2 | 30% | 3% | | 3 | 20% | 5% | An item is selected at random from today's output and found to be defective. (a) What is the posterior probability that the item came from each production line? (b) A second item is now drawn independently from the same (unknown) production line. What is the conditional probability that this second item is also defective, given that the first was defective?概率中等数值题未尝试免费045Multi-Analyst Signal Fusion and Sequential UpdatingA trading desk receives directional predictions from three independent analysts. The market will either go "up" or "down", each with prior probability 1 2 . Each analyst independently (given the true state) predicts the correct direction with the following probabilities: - Analyst 1: accuracy 0.8 - Analyst 2: accuracy 0.7 - Analyst 3: accuracy 0.9 (a) All three analysts predict "up". What is P( up \mid all three say up )? (b) Analysts 1 and 2 predict "up", but Analyst 3 predicts "down". What is P( up \mid U 1, U 2, D 3)? (c) Show that the posterior in part (b) can be computed by sequential Bayesian updating — updating on one analyst's signal at a time — and that the final answer does not depend on the order of updates. Demonstrate this by computing the posterior via two different orderings.概率困难derivation未尝试面试订阅059All Triples Independent but Quadruple NotLet \Omega = \ 0, 1, 2, \ldots, 7\ with uniform probability P(\ \omega\ ) = 1/8. Write each \omega in binary as (b 2, b 1, b 0). Define events: A = \ \omega : b 0 = 1\ , \quad B = \ \omega : b 1 = 1\ , \quad C = \ \omega : b 2 = 1\ , \quad D = \ \omega : b 0 \oplus b 1 \oplus b 2 = 1\ . (a) Show that A, B, C are mutually independent. (b) Show that any triple chosen from \ A,B,C,D\ is mutually independent. (c) Show that A, B, C, D are not mutually independent.概率困难derivation未尝试面试订阅065Mixtures of Independent Distributions Destroy IndependenceA biased coin lands heads with probability 1/2. If heads, set (X, Y) = (1, 1). If tails, draw X and Y independently, each Bernoulli (1/2). (a) Compute the full joint distribution of (X, Y). (b) Show that X and Y have the same marginal distribution. (c) Are X and Y independent? Prove your answer. (d) Compute P(X = 1 \mid Y = 0) and compare it to P(X = 1). Interpret the result.概率困难derivation未尝试面试订阅088The St. Petersburg ParadoxA casino offers the following game. A fair coin is flipped repeatedly until the first Tails appears. If the first Tails occurs on flip n, you win 2 n dollars. (a) Compute the expected payoff of the game. (b) Despite the answer to (a), most people would pay no more than about \20 to play. Resolve this apparent paradox using Daniel Bernoulli's approach: assume the player has log utility u(x) = \ln(x) and initial wealth W. Compute the expected utility of the game and find the certainty equivalent (the sure amount that gives the same expected utility) for W = 1 , 000 , 000. (c) A more practical resolution: suppose the casino has finite total capital C. If the payout is capped at C = 2 40 (about \1 trillion), what is the expected payoff?概率中等derivation未尝试免费090Borel's Paradox: Conditioning on Measure-Zero EventsLet (\Theta, \Phi) be uniformly distributed on the unit sphere S 2, where \Theta \in [0, 2 ) is the longitude and \Phi \in [0, ] is the colatitude, with joint density f( , \phi) = 1 4 \sin \phi. (a) Compute the conditional distribution of \Phi given \Theta = 0 by treating \Theta as the conditioning variable (i.e., compute f(\phi \mid = 0)). (b) Now reparameterize: let X = \cos(\Theta) \sin(\Phi), Y = \sin(\Theta) \sin(\Phi), Z = \cos(\Phi). The great circle \ \Theta = 0\ can equivalently be described as \ Y = 0, X 0\ . Compute the conditional distribution of \Phi given Y = 0 and X > 0. Is it the same as in part (a)? (c) Explain why the two answers differ. What does this tell us about the meaning of 'conditioning on a measure-zero event,' and what is the correct mathematical framework for resolving this ambiguity?概率困难derivation未尝试免费164Exactly One Pair Among Four BirthdaysFour people have independent uniform birthdays on a 365-day calendar. What is the probability that there is exactly one matching pair and no larger collision?概率困难derivation未尝试面试订阅170Collision Risk for the Next Arrival Given Distinct Existing BirthdaysSuppose n existing birthdays are all distinct on a 365-day calendar. A new person's birthday is then drawn uniformly and independently. What is the probability the new arrival creates an exact collision?概率困难derivation未尝试面试订阅190Expected Number of Odd-Load BinsEight labeled balls are independently assigned to five labeled bins. What is the expected number of bins that end up with an odd load?概率困难数值题未尝试免费200Full Distribution of Empty Urns via Stirling NumbersSix distinguishable balls are thrown independently and uniformly at random into 5 distinguishable urns. Let E denote the number of empty urns. Derive the probability mass function P(E = k) for every possible value of k, expressing each probability as an exact fraction.概率困难derivation未尝试免费217Negative Binomial as a Poisson–Gamma MixtureLet \Lambda \sim Gamma (r, ) with density f \Lambda( ) = r \Gamma(r) r-1 e - for > 0, and let X \mid \Lambda = \sim Poisson ( ). (a) Write down P(X = k \mid \Lambda = ) and compute the marginal PMF P(X = k) by integrating over \Lambda. (b) Show that P(X = k) = \binom k + r - 1 k p k (1-p) r where p = 1 1+ , and identify the distribution. (c) Use the mixture representation to find E[X] and Var (X) via the tower property (law of total expectation and law of total variance), without computing the PMF. (d) Verify numerically: r = 3, = 4. Compute P(X = 2) and E[X].概率中等derivation未尝试免费220Poisson Limit of the Binomial via Characteristic FunctionsLet X n \sim Binomial (n, /n) for fixed > 0 and n > . (a) Write down the characteristic function \varphi X n (t) = E[e itX n ] in closed form. (b) Show that \lim n \varphi X n (t) = e (e it - 1) for every t \in R . (c) Identify the limiting characteristic function and state the convergence-in-distribution conclusion. (d) Justify why pointwise convergence of characteristic functions implies convergence in distribution (cite the relevant theorem). (e) For = 5, n = 100: compute P(X n = 3) using both the exact Binomial PMF and the Poisson approximation, and find the relative error.概率困难derivation未尝试免费285Robust Monochromatic Cliques in a Random Edge-ColoringEach edge of the complete graph K n is independently colored red or blue with equal probability 1 2 . For a fixed integer k \ge 2, find the expected number of monochromatic k-cliques (complete subgraphs on k vertices whose edges are all the same color). Express your answer in terms of n and k. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费288Robust Isolated Vertices in a Random GraphIn the Erdos-Renyi random graph model G(n,p), each of the \binom n 2 possible edges among n labeled vertices is included independently with probability p. A vertex is isolated if it has no edges. Find the expected number of isolated vertices. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费295Robust Cycles in a Random PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . Find the expected number of cycles in the cycle decomposition of . Express your answer as a familiar function of n. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费300Robust Common Edges of Two Random Spanning TreesLet T 1 and T 2 be two independent uniformly random spanning trees of the complete graph K n (each drawn uniformly at random from all n n-2 labeled spanning trees, independently of the other). Find the expected number of edges that belong to both T 1 and T 2. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费303Robust Coincident RollsYou roll a fair k-sided die n times independently. Let M be the number of pairs (i, j) with 1 \le i < j \le n such that roll i equals roll j. Find E[M]. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费