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427Conditional Expectation via MemorylessnessLet X \sim Exp (2). Using the memoryless property, compute E[X \mid X > 3].概率简单数值题未尝试免费429Geometric Number of Geometric TrialsA gambler plays a sequence of rounds. In each round, he flips a biased coin with P( heads ) = p repeatedly until he gets heads; the number of flips in that round is Geom (p). The number of rounds he plays is itself Geom (q) (independent of the coin flips), where 0 < q < 1. Let S be the total number of coin flips across all rounds. Using the memoryless property of the geometric distribution, show that S \sim Geom (pq) and compute E[S].概率中等derivation未尝试免费430Characterization of Memorylessness and the Residual Life ParadoxPart (a): Let X be a continuous, positive random variable satisfying P(X > s + t \mid X > s) = P(X > t) for all s, t 0. Prove that X must be exponentially distributed. Part (b): A lightbulb's lifetime L has CDF F(t) = 1 - 1 2 e -t - 1 2 e -3t for t 0 (a mixture of Exp (1) and Exp (3)). You arrive at a uniformly random time and observe the bulb currently in use. Let R be the residual lifetime of that bulb. Show that E[R] > E[L] and compute both values. Explain why memorylessness breaks down and causes this paradox.概率困难derivation未尝试面试订阅431Geometric Survival Past a ThresholdLet X \sim Geom (1/4) (number of trials until first success). Using the memoryless property of the geometric distribution, compute (i) E[X \mid X > 5] and (ii) P(X > 8 \mid X > 5).概率简单数值题未尝试免费432Asymmetric Penalties in an Exponential RaceTwo independent alarms go off at Exp (4) and Exp (6) times respectively. If alarm 1 fires first you pay \3; if alarm 2 fires first you pay \5. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an additional \1 when it fires. Find the expected total payment.概率中等数值题未尝试免费433Conditional Variance of a Surviving ExponentialLet X \sim Exp ( ). Using the memoryless property, find Var (X \mid X > t) for t > 0. Does conditioning on survival change the variance compared to Var (X)? Evaluate numerically for = 5 and t = 2.概率中等数值题未尝试免费434Second Failure in a Memoryless Component ArrayA system has 4 independent components, each with lifetime Exp (2). When a component fails, it is removed and the remaining components continue operating. By memorylessness, surviving components' residual lifetimes are still Exp (2). Find the expected time until the second component fails.概率中等数值题未尝试免费435Uniqueness of Geometric MemorylessnessPart (a): Let N be a positive-integer-valued random variable satisfying P(N > m + n \mid N > m) = P(N > n) for all m, n \in Z 0 . Prove that N must follow a geometric distribution. Part (b): For N \sim Geom (p), compute E[N 2 \mid N > k] using memorylessness and verify that Var (N \mid N > k) = Var (N).概率困难derivation未尝试面试订阅436Direct Application of Exponential MemorylessnessA radioactive atom has a lifetime X \sim Exp (1/2). Given that the atom has survived past time t = 3, what is the probability it survives past time t = 7?概率简单数值题未尝试免费437Fresh Start After a Losing StreakYou flip a coin with P( heads ) = 1/3 until you get heads. Given that the first 8 flips were all tails, what is the expected total number of additional flips needed (starting from flip 9)?概率简单数值题未尝试免费438Machine Replacements via Memoryless MinimumA factory runs 3 identical machines with independent lifetimes Exp (1). When any machine fails, it is instantly replaced with a new identical machine. All non-failed machines continue running (their residual lifetimes remain Exp (1) by memorylessness). Find the expected number of machine replacements in the time interval [0, 10].概率中等数值题未尝试免费439Sequential Elimination RaceThree players with independent lifetimes X 1 \sim Exp (1), X 2 \sim Exp (2), X 3 \sim Exp (4) compete. The first to "die" is eliminated, then the two survivors continue (by memorylessness, their residual lifetimes are fresh exponentials with the same rates). The second to die is eliminated, and the last survivor wins. (a) Find the probability that the elimination order is X 3, X 1, X 2 (i.e., player 3 dies first, then player 1, then player 2). (b) Find the expected total time until only one player remains.概率困难multi part未尝试面试订阅442Constant Hazard Rate from MemorylessnessA device's lifetime X has survival function F (t) = P(X > t) and hazard rate h(t) = f(t)/ F (t). Show that the memoryless property P(X > s + t \mid X > s) = P(X > t) implies h(t) = (a constant) for all t 0, and conversely that a constant hazard rate implies the memoryless property. Conclude that X \sim Exp ( ).概率中等derivation未尝试免费443Series System Replacement Costs via Competing ExponentialsA machine has two critical components in series: component A with lifetime Exp (3) and component B with lifetime Exp (5), independent of each other. When either fails, the entire machine stops, the failed component is replaced (cost \20 for A, \50 for B), and both components restart fresh (the survivor restarts by memorylessness, the replacement is new). Find the expected replacement cost per unit time in the long run.概率中等数值题未尝试免费444Full Ordering Probability for Four Competing ExponentialsFour independent exponential random variables X 1 \sim Exp (1), X 2 \sim Exp (2), X 3 \sim Exp (3), X 4 \sim Exp (6) represent task completion times. Using iterated applications of the memoryless property, find P(X 4 < X 3 < X 2 < X 1) — the probability that the tasks complete in the specific order 4, 3, 2, 1.概率困难数值题未尝试面试订阅445Memorylessness Breaks for Exponential MixturesLet X have the mixture density f(x) = 1 2 e -x + 5 2 e -5x for x 0 (a 50 - 50 mixture of Exp (1) and Exp (5)). (a) Compute P(X > s + t \mid X > s) as a function of s and t, and show it depends on s (i.e., the memoryless property fails). (b) Evaluate P(X > 2 \mid X > 1) and compare with P(X > 1). (c) Interpret: given that X has survived past s, how does the conditional distribution change as s increases?概率困难multi part未尝试面试订阅446Geometric Coupon Collector's Conditional SurvivalYou roll a fair die repeatedly until you roll a 6. Let N be the number of rolls needed. Given that you have not yet rolled a 6 in the first 5 rolls, what is the probability that you will need more than 10 rolls total?概率简单数值题未尝试免费447The Memoryless BusA bus arrives at a stop at an Exp (1/10) random time (mean 10 minutes). You have already been waiting for 5 minutes. What is the expected additional waiting time?概率简单数值题未尝试免费448Threshold Exceedance for the Minimum of Two ExponentialsLet X \sim Exp (2) and Y \sim Exp (3) be independent. Define M = \min(X, Y) and fix a threshold c = 1. (i) Find P(M > 1). (ii) Given M > 1, find E[M - 1 \mid M > 1] and P(X < Y \mid M > 1) (the probability that X is the minimum, given both survived past 1).概率中等数值题未尝试免费449Memoryless Message Relay ChainA message must traverse a chain of relay nodes to reach its destination. Each node independently takes Geom (1/3) attempts to successfully forward the message to the next node. However, on each attempt, there is an independent probability 1/5 that the node permanently fails, destroying the message. If the node fails, the message is lost. If the chain has 2 nodes, find: (i) The probability the message reaches the destination (traverses both nodes). (ii) The expected total number of attempts across both nodes, given the message reaches the destination.概率中等数值题未尝试免费