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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5981Total Stake Until the First SixYou roll a fair die repeatedly until the first time a 6 appears; let N be the number of rolls (including the winning roll). On each roll you independently collect a payoff X i with E[X i]=1.5, where the X i are i.i.d. and independent of the roll values. Compute E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5982Variance of Fills Over a Binomial Number of QuotesOut of n=10 resting quotes, each fills independently with probability 0.3, so the number of fills N is Binomial(10,0.3). Each fill produces an i.i.d. PnL X i with E[X i]=2 and Var (X i)=9, independent of which quotes fill. For the stopped sum S N=\sum i=1 N X i, compute Var (S N).概率中等derivation未尝试免费5983Number of Trades to Cross a Profit TargetA strategy books i.i.d. positive profits X 1,X 2,\dots with E[X i]=2.5. Let N be the first time the running total S n=\sum i=1 n X i strictly exceeds 10; that is, N=\min\ n:S n>10\ . Assuming E[N]< , the expected overshoot is known to satisfy E[S N]=14. Use a Wald-style identity to compute E[N].概率中等derivation未尝试免费5984Expected Inspection Cost Until the First DefectA quality line inspects items one at a time; each item is defective independently with probability 0.05. Inspection stops at the first defective item. Each inspection (defective or not) costs an i.i.d. amount C i with E[C i]=\8, independent of the defect outcomes. Let N be the number of items inspected. Find the expected total inspection cost E\! [\sum i=1 N C i ].概率简单数值题未尝试免费5985Expected Total Slippage With Negative DriftA market-making desk incurs i.i.d. per-trade adverse-selection costs X 1,X 2,\dots with E[X i]=-0.4 (a net loss per trade). The number of trades in a session, N, is independent of the costs and is Poisson with mean 15. Compute the expected cumulative cost E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5986Expected Winnings Over a Random Number of BetsA gambler places bets until a random stopping rule halts play; the number of bets N is a stopping time for the i.i.d. bet outcomes with E[N]=8. Each bet has an i.i.d. net result X i with E[X i]=-0.05 (a 5\% house edge per unit staked, with unit stakes), and the decision to stop after bet n depends only on outcomes up to bet n. Compute the gambler's expected total winnings E\! [\sum i=1 N X i ], and state whether any stopping rule with E[N]=8 can make this positive.概率中等derivation未尝试免费5987When the Stopping Rule Looks at the Last DrawDraw i.i.d. values X 1,X 2,\dots uniform on \ 1,2,3\ (so E[X i]=2). Define N as follows: keep drawing and stop the first time you draw a 3; let N be the number of draws. Let S N=\sum i=1 N X i. A candidate computes E[N]E[X 1]=3 2=6 and claims E[S N]=6. Compute the correct value of E[S N] and explain in one sentence why E[N]E[X 1] is the wrong formula here.概率困难essay未尝试面试订阅5988Expected Sample Size of a Sequential Drift TestA sequential test accumulates i.i.d. log-likelihood increments X 1,X 2,\dots with E[X i]=0.25. The test stops at N=\min\ n: |S n|\ge 3\ where S n=\sum i=1 n X i, and it is given that E[N]< and that the expected stopped statistic is E[S N]=2.0 (reflecting that the upper boundary is hit far more often under this positive drift). Each observation costs \6 to collect. Using a Wald-style identity, find the expected total data-collection cost.概率中等数值题未尝试免费