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3063At Least One Arrival in the Final Quarter Given Four TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 4 arrivals in the hour, what is the probability that at least one arrival falls in the final quarter-hour?概率中等derivation未尝试面试订阅3064Expected Idle Time Before the First Arrival Given Three TotalA Poisson process is observed on [0,1] hour. Conditional on exactly 3 arrivals in the hour, what is the expected idle time from 0 until the first arrival?概率中等derivation未尝试面试订阅3065Expected Idle Time After the Last Arrival Given Five Total in Two HoursA Poisson process is observed on [0,2] hours. Conditional on exactly 5 arrivals in the two-hour horizon, what is the expected idle time from the last arrival until hour 2?概率中等derivation未尝试面试订阅3166Signal Before a Binary TradeA trade pays +8 in a favorable state and -5 in an unfavorable state. The favorable state has prior probability 2 5 . Before trading, you may buy a signal for cost 1 2 ; it is correct with probability 4 5 . If you see the signal, you may either trade or abstain after observing it. What is the value of the signal, and should you buy it at that cost?概率中等derivation未尝试面试订阅3176Aggressive vs Defensive Quote After a SignalThere are two possible actions. `Aggressive` pays 10 in the good state and -8 in the bad state. `Defensive` pays 4 in the good state and -1 in the bad state. The good state has prior probability 2 5 . Before acting, you may see a binary signal that is correct with probability 4 5 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3177Allocate Between Fast and Safe BooksThere are two possible actions. `Aggressive` pays 9 in the good state and -6 in the bad state. `Defensive` pays 5 in the good state and 1 in the bad state. The good state has prior probability 1 2 . Before acting, you may see a binary signal that is correct with probability 3 4 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3186Perfect Information Before Choosing a Desk StrategyTwo actions are available before the state is revealed. Action A pays 10 in the good state and -4 in the bad state. Action B pays 4 in the good state and 3 in the bad state. The good state has prior probability 2 5 . What is the expected value of perfect information about the state before acting?概率中等derivation未尝试面试订阅3191Total PnL Until a Geometric Number of FillsLet X 1,X 2,\dots be i.i.d. increments with E[X i]=3 and Var (X i)=5. Let N be independent of the increments and distributed as Geometric( 1 4 ) on 1,2,\dots . For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3192Aggregate Slippage Over a Poisson Number of OrdersLet X 1,X 2,\dots be i.i.d. increments with E[X i]=2 and Var (X i)=3. Let N be independent of the increments and distributed as Poisson(4). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3193Total Cost Over a Negative-Binomial HorizonLet X 1,X 2,\dots be i.i.d. increments with E[X i]=4 and Var (X i)=6. Let N be independent of the increments and distributed as NegativeBinomial(r=3, p= 2 5 ). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3201Expected Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style reasoning to compute E[T].概率中等derivation未尝试面试订阅3206Variance of Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style second-moment reasoning to compute Var (T).概率困难derivation未尝试面试订阅3212Second Moment of Centered Sum at a Poisson HorizonLet X 1,X 2,\dots be i.i.d. with mean and variance 3. Let N be independent of the increments and distributed as Poisson(4). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation未尝试面试订阅3214Centered Slippage Variance Under Random StoppingLet X 1,X 2,\dots be i.i.d. with mean and variance 4. Let N be independent of the increments and distributed as Geometric( 1 3 ). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation未尝试面试订阅5893Deriving the Even-Money Kelly FractionYou repeatedly bet a fraction f of your current wealth on an even-money wager that wins with probability p>\tfrac12 (you gain the staked amount on a win, lose it on a loss). By maximizing the expected logarithm of your wealth multiplier over one round, derive the growth-optimal fraction f *.概率简单derivation未尝试免费5894Kelly Fraction at General Net OddsA favorable bet pays net odds b to 1: staking an amount, you gain b times the stake with probability p and lose the stake with probability 1-p. Betting a fraction f of wealth each round, derive the growth-optimal fraction f * in terms of b and p.概率简单derivation未尝试免费5895Maximum Growth Rate of a Kelly BettorAn even-money coin wins with probability p=0.6. You bet the growth-optimal (Kelly) fraction every round. Compute the resulting maximum expected log-growth rate per round, and express it in closed form in terms of p.概率中等数值题未尝试免费5896Why Half-Kelly Keeps Three-Quarters of the GrowthFor a small-edge repeated bet the expected log-growth is well approximated by the quadratic G(f)\approx f-\tfrac12 2 f 2, where and 2 are the per-round mean and variance of the bet's return. Using this approximation, find the optimal fraction f * and show what fraction of the maximal growth G(f *) is retained by betting half-Kelly, f=f */2.概率中等derivation未尝试面试订阅5897Overbetting to Twice KellyUnder the small-edge approximation G(f)\approx f-\tfrac12 2 f 2 for the expected log-growth of a repeated bet, the growth-optimal fraction is f *= / 2. At what (nonzero) betting fraction does the expected log-growth fall back to zero, and what does this say about the symmetry of growth around f *?概率中等数值题未尝试面试订阅5898Continuous Kelly for Normal ReturnsEach round you allocate a fraction f of wealth to a position whose one-period return R is approximately normal with small mean >0 and variance 2 (with 2\ll 2), so post-round wealth is multiplied by 1+fR. Using a second-order expansion of the log, derive the growth-optimal fraction f *.概率中等derivation未尝试面试订阅