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5953Collecting in PairsEach purchase gives you a PACK of 2 cards, where each of the 2 cards is independently and uniformly one of 4 types (the two cards in a pack may coincide). What is the expected number of PACKS you must buy to collect all 4 types?概率困难数值题未尝试面试订阅5954Just These TwoCoupons arrive uniformly and independently from 5 types. You only care about two SPECIFIC types, the gold and the silver coupon; the other three are worthless to you. What is the expected number of draws until you have collected BOTH the gold and the silver coupon?概率中等数值题未尝试免费5955Buy Random or Buy the Missing OneYou need all 5 types and currently hold 4 distinct types (exactly one type missing). Each round you may either (a) buy a random coupon for 1 (uniform over all 5 types), or (b) directly buy your missing type from a reseller for 5. Acting optimally to minimize expected total future cost, what is your minimum expected cost to complete the set?概率中等数值题未尝试免费5957The Busiest Bin3 balls are thrown independently and uniformly into 3 bins. Let M be the maximum load, i.e. the number of balls in the most-loaded bin. What is E[M]?概率中等数值题未尝试免费5959Coupons With BlanksEach draw is a blank with probability 1/4 (no coupon), and otherwise (probability 3/4) yields one of 3 collectible types, each equally likely. What is the expected number of draws to collect all 3 types?概率中等数值题未尝试免费5960Every Bin Gets a PairBalls are dropped one at a time, each independently into one of 2 equally likely bins. What is the expected number of balls dropped until EVERY bin contains at least 2 balls?概率中等数值题未尝试免费5972Hitting the Opposite Node on a CycleA token does a symmetric random walk on the 6 vertices of a cycle (labelled 0..5, each step moving to one of the two adjacent vertices with probability 1/2 each). Starting at vertex 0, find the expected number of steps to first reach vertex 3, the diametrically opposite vertex.概率简单数值题未尝试免费6024Persistence and the Covariance-Stationarity VerdictA GARCH(1,1) has =0.20, =0.75. Compute the persistence + and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.统计简单derivation未尝试面试订阅6029News-Impact Update from a Signed ReturnA GARCH(1,1) has \omega=0.00001, =0.08, =0.90. Today's conditional variance is h t=0.0004 and today's return is r t=-0.03. Compute tomorrow's conditional variance h t+1 as a decimal.统计中等derivation未尝试面试订阅6051Martingale Condition for W Squared Minus c tLet W t be standard Brownian motion. For what constant c is the process M t = W t 2 - c t a martingale?随机过程中等数值题未尝试免费6052Direction of Mean Reversion in a CIR ProcessA CIR process satisfies dX t = 2(0.04 - X t) dt + 0.1 sqrt(X t) dW t. The current value is X t = 0.07. Is the instantaneous drift positive or negative, and what is its numerical value?随机过程简单数值题未尝试免费