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049Mystery Coin IdentificationA box contains three coins, each equally likely to be selected: - **Coin F** (fair): P(H) = 1 2 - **Coin H** (heads-biased): P(H) = 3 4 - **Coin T** (tails-biased): P(H) = 1 4 You pick one coin uniformly at random and flip it three times, obtaining the sequence H, T, H (flips are conditionally independent given the coin). (a) Find the posterior probability that the coin is each type. (b) What is the conditional probability that the fourth flip is heads?概率困难数值题未尝试免费050Noisy Signal Detection and Evidence ThresholdA hidden signal S is equally likely to be +1 or -1. At each time step you receive a noisy reading: if S = +1 the reading is +1 with probability 2 3 and -1 with probability 1 3 ; if S = -1 the reading is -1 with probability 2 3 and +1 with probability 1 3 . Readings are conditionally independent given S. (a) You observe the sequence (+1, +1, -1). Find the posterior probability P(S = +1 \mid observations ). (b) Starting from the uniform prior, what is the minimum number n of consecutive +1 readings required so that P(S = +1 \mid n consecutive +1) > 0.95?概率困难数值题未尝试免费078Simpson's Paradox in a Clinical TrialA clinical trial tests a drug on two subgroups. In subgroup A (mild cases): 81/87 (93%) of treated patients recover vs 234/270 (87%) of untreated. In subgroup B (severe cases): 192/263 (73%) of treated patients recover vs 55/80 (69%) of untreated. The drug improves recovery rates in both subgroups. Now compute the overall recovery rates (combining subgroups) for treated vs untreated. Explain the apparent contradiction and identify the lurking variable that causes it.概率中等derivation未尝试免费080The Two-Envelope ParadoxTwo envelopes each contain a positive amount of money; one contains exactly twice the other. You pick one envelope at random and find it contains x dollars. The naive argument says: the other envelope is equally likely to contain 2x or x/2, so the expected value of switching is (1/2)(2x) + (1/2)(x/2) = 5x/4 > x, and you should always switch — but this leads to the absurd conclusion that you should switch back and forth indefinitely. (a) Identify the precise flaw in the naive argument. (b) Suppose the smaller amount S is drawn from a known proper prior distribution with E[S] = < . Show that the unconditional expected gain from switching is zero. (c) Explain why conditional on observing x, it *can* be rational to switch for some values of x and not others.概率困难derivation未尝试面试订阅082The Sleeping Beauty ProblemSleeping Beauty participates in the following experiment. On Sunday she is put to sleep. A fair coin is flipped. If it lands Heads, she is woken on Monday only. If it lands Tails, she is woken on Monday and again on Tuesday (with her memory of the Monday awakening erased before Tuesday). Each time she wakes, she is asked: 'What is your credence that the coin landed Heads?' She knows the full protocol. (a) Present the argument that her answer should be 1/3 (the 'thirder' position). (b) Present the argument that her answer should be 1/2 (the 'halfer' position). (c) Suppose the experiment is repeated 1000 times independently. If Beauty bets \1 at even odds on Heads each time she is woken, what is her expected net gain or loss over all awakenings? What does this imply about the two positions?概率中等derivation未尝试免费083The Necktie ParadoxTwo players, Alice and Bob, each receive a necktie as a gift. The prices of the two neckties are different positive values. Neither player knows either price. They agree to compare: whoever has the cheaper necktie wins the other's necktie. Alice reasons: 'If my necktie costs x, then I either gain a necktie worth more than x or lose a necktie worth x. Since each is equally likely, my expected gain from the game is positive.' Bob makes the identical argument. Both conclude the game favors them — a contradiction since this is a zero-sum exchange. (a) Identify the precise flaw in Alice's reasoning. (b) Suppose the two necktie prices are drawn by a third party as V and 2V where V \sim Uniform (1, 100), assigned randomly to Alice and Bob. If Alice sees her necktie has price tag x, what is her expected gain from the game as a function of x? Show that the unconditional expected gain is zero.概率中等derivation未尝试免费085The Inspection Paradox (Bus Waiting Time)Buses arrive at a stop according to a Poisson process with rate (so inter-arrival times are iid Exp ( ) with mean 1/ ). You arrive at the bus stop at a uniformly random time, independent of the bus schedule. Let L be the length of the inter-arrival interval that contains your arrival time — i.e., the time between the last bus before you arrived and the next bus after. (a) Find E[L]. Explain why it is **not** 1/ despite inter-arrival times having mean 1/ . (b) Find the expected waiting time E[W] until the next bus, where W is the time from your arrival until the next bus. (c) A city official surveys bus riders and asks how long they waited. If the reported average is 1/ , should the transit authority be surprised? Explain using the inspection paradox.概率困难derivation未尝试面试订阅087The Gambler's Fallacy and the Hot-Hand FallacyA fair coin is flipped 100 times. (a) Conditional on the coin having just landed Heads three times in a row, what is the probability the next flip is Heads? A gambler insists it should be less than 1/2 because the coin 'is due for Tails.' Explain the error precisely. (b) Now consider a different setting: a basketball player's shots are modeled as a Markov chain where P( Hit \mid previous Hit ) = 0.6 and P( Hit \mid previous Miss ) = 0.4. After three consecutive hits, what is the probability the next shot is a hit? (c) Miller and Sanjurjo (2018) showed that even with a fair coin, if you select flips that follow a streak of k Heads and compute the sample proportion of Heads among those selected flips, the expected proportion is strictly less than 1/2. Why does this happen, and what does it imply for empirical hot-hand studies?概率简单derivation未尝试免费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089The Prosecutor's FallacyA city has 1,000,000 residents. A crime is committed by one person. The police use a forensic test that has a false positive rate of 1 in 10,000 (i.e., P( positive \mid innocent ) = 10 -4 ) and a true positive rate of 100% (P( positive \mid guilty ) = 1). A suspect tests positive. (a) The prosecutor argues: 'The probability of a false positive is only 10 -4 , so there is only a 0.01% chance this person is innocent.' What is wrong with this argument? (b) Compute the actual probability the suspect is guilty given the positive test, assuming the suspect was selected uniformly at random from the city before the test. (c) Now suppose eyewitness evidence independently narrows the suspect pool to 5 people before the forensic test is applied. Recompute the posterior probability of guilt.概率中等数值题未尝试免费098The Will Rogers Paradox (Stage Migration)Group A consists of the values \ 1, 2, 3, 9\ and Group B consists of \ 5, 6, 7, 8\ . (a) Compute the mean of each group. (b) Now move the element 5 from Group B to Group A. Recompute the mean of each group. Verify that both means have increased. (c) The overall mean across all 8 values is unchanged. Prove the following general statement: if an element is moved from group B to group A, and the element's value x satisfies A < x < B (where A , B are the pre-move group means), then both post-move group means strictly exceed their pre-move values. (d) A medical study reports that a new diagnostic technology has improved average survival time in both Stage I and Stage II cancer patients, yet overall survival is unchanged. Explain how this is possible without any actual improvement in treatment.概率中等数值题未尝试免费135Conditional Third Roll Reaching TenYou roll two fair six-sided dice. If their sum is even, you roll a third die and your score is the sum of all three dice. If their sum is odd, your score is just the sum of the first two dice. What is the probability that your score is at least 10?概率困难数值题未尝试免费167Expected Days Hit by Both GroupsGroup A has a people and group B has b people, with independent uniform birthdays on a 365-day calendar. What is the expected number of calendar days on which both groups have at least one birthday?概率中等derivation未尝试面试订阅180Hypergeometric Moments from an UrnAn urn contains 20 balls: 8 red and 12 blue. You draw 5 balls without replacement. Let X be the number of red balls drawn. Derive E[X] and Var (X) using indicator random variables. Express each answer as an exact fraction.概率困难derivation未尝试免费193Conditional Expected Occupancy of a Nonempty UrnFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Given that urn 1 is nonempty, what is the expected number of balls in urn 1? Give an exact fraction.概率中等数值题未尝试免费210Multinomial Covariance and Conditional DistributionA fair die is rolled n = 60 times. Let X i be the number of times face i appears, for i = 1, \ldots, 6, so (X 1, \ldots, X 6) \sim Multinomial (60,\, 1/6, \ldots, 1/6). (a) Using indicator variables, compute Cov (X 1, X 2). (b) Find the correlation (X 1, X 2). (c) Determine the conditional distribution of (X 2, X 3, X 4, X 5, X 6) given X 1 = 12. What is E[X 2 \mid X 1 = 12]?概率困难derivation未尝试免费214Poisson Thinning and Independence of Split StreamsLet N \sim Poisson ( ). Each of the N events is independently classified as type 1 with probability p and type 2 with probability 1 - p. Let N 1 and N 2 denote the counts of type 1 and type 2 events, respectively. (a) Derive the marginal distribution of N 1. (b) Derive the joint PMF P(N 1 = j, N 2 = k) and show that N 1 and N 2 are independent. (c) A website receives page views at rate = 200 per hour. Each visitor independently converts (makes a purchase) with probability p = 0.03. Find the probability of exactly 4 conversions in an hour, and the probability of at least 1 conversion given at most 210 total page views.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223Zero-Truncated Poisson DistributionThe **zero-truncated Poisson** distribution arises when a Poisson process is observed conditional on at least one event occurring. Let Y \sim Poisson ( ) with > 0, and define X = (Y \mid Y \ge 1). (a) Derive the PMF of X: show that P(X = k) = k k!(e - 1) for k = 1, 2, 3, \ldots, and verify it sums to 1. (b) Compute E[X] by relating it to E[Y] via the truncation. Specifically, show that E[X] = 1 - e - . (c) Derive Var (X) using the identity Var (X) = E[X 2] - (E[X]) 2. Show that Var (X) = (1 + ) 1 - e - - 2 (1 - e - ) 2 . (d) Evaluate P(X = 1), E[X], and Var (X) numerically for = 0.5.概率中等derivation未尝试免费224Compound Poisson Distribution: MGF and MomentsLet N \sim Poisson ( ) and let X 1, X 2, \ldots be iid discrete random variables (independent of N) with PMF P(X i = j) = p j for j = 1, 2, \ldots and MGF M X(t) = E[e tX 1 ]. Define the **compound Poisson** sum S = \sum i=1 N X i (with S = 0 when N = 0). (a) Derive the MGF of S. Show that M S(t) = \exp\!\big( (M X(t) - 1)\big). (b) Use the MGF to derive E[S] and Var (S). Express your answers in terms of , E[X 1], and Var (X 1). (c) Alternatively, derive E[S] and Var (S) using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on N. (d) **Application:** An insurance company receives claims at a Poisson rate of = 10 per day. Each claim size is \1000 with probability 0.6 or \5000 with probability 0.4, independently. Find E[S] and Var (S) for the total daily claims S, and compute the standard deviation.概率困难derivation未尝试免费