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030Three Suppliers and a Two-Stage InspectionA firm sources components from three suppliers: S 1 (50% of supply, 2% defect rate), S 2 (30%, 3% defect rate), S 3 (20%, 5% defect rate). A component is selected at random and undergoes two independent inspection stages. Stage 1 detects a defect with probability 0.8 (if defective) and falsely flags a good component with probability 0.05. Stage 2 detects a defect with probability 0.9 (if defective) and falsely flags with probability 0.03. A component is flagged by both stages. Compute: (a) P( component is actually defective \mid flagged by both stages ). (b) Given that the component is actually defective and flagged by both stages, what is the probability it came from S 3?概率困难数值题未尝试面试订阅084Nontransitive DiceThree dice have the following faces. Die A: 2, 2, 4, 4, 9, 9 . Die B: 1, 1, 6, 6, 8, 8 . Die C: 3, 3, 5, 5, 7, 7 . Each die is fair (all six faces equally likely). Two players each choose a die and roll it; the higher number wins. (a) Compute P(A > B), P(B > C), and P(C > A). (b) Show that these dice are nontransitive: A tends to beat B, B tends to beat C, yet C tends to beat A. (c) You are in a game where your opponent chooses a die first, then you choose. Which die should you pick in each case, and what is your winning probability?概率中等数值题未尝试免费356Tower Property with a Three-Level Discrete LatentA random variable K is drawn uniformly from \ 1, 2, 3\ . Given K = k, the random variable X \sim Exp (k) (rate k, so E[X \mid K = k] = 1/k). Find E[X].概率简单数值题未尝试免费361Random Number of Coin Flips via Tower PropertyA fair die is rolled to obtain D \sim Uniform \ 1,2,3,4,5,6\ . Then D independent fair coins are flipped and X equals the total number of heads. Using the tower property, find E[X].概率简单数值题未尝试免费362Two-Stage Binomial Draw via Iterated ExpectationLet N be drawn uniformly from \ 1, 2, 3, 4\ , and given N = n, let X \sim Binomial (n, 1/3). Find E[X].概率简单数值题未尝试免费366Product Moment via Tower Conditioning on One FactorLet Y \sim Exp (1) and, given Y = y, let X \mid Y = y \sim Uniform (0, y). Using the tower property, compute E[XY].概率简单数值题未尝试免费371Beta-Uniform Prior on Binomial Success ProbabilityLet P \sim Uniform (0,1) and, given P = p, let X \mid P = p \sim Binomial (10, p). Using the tower property, find E[X].概率简单数值题未尝试免费372Expected Maximum of Correlated Bernoullis via Indicator and TowerLet U \sim Uniform (0,1) and, given U, let X and Y be conditionally i.i.d.\ Bernoulli (U). Define M = \max(X, Y). Using the tower property and the indicator representation M = 1 \ X \ge 1 or Y \ge 1\ , find E[M].概率简单数值题未尝试免费378Distribution of the Sum of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the convolution formula, derive the PDF of Z = X + Y.概率中等derivation未尝试免费384Distribution of the Product of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the transformation (W, V) = (XY,\, Y), derive the PDF of W = XY.概率中等derivation未尝试免费477Mean Return Time and the Stationary DistributionA Markov chain on \ 1, 2, 3\ has transition matrix P = \begin pmatrix 0 & \tfrac 1 2 & \tfrac 1 2 \\ \tfrac 1 4 & \tfrac 1 2 & \tfrac 1 4 \\ \tfrac 1 3 & \tfrac 1 3 & \tfrac 1 3 \end pmatrix . **(a)** Find the stationary distribution . **(b)** Using the relationship between the stationary distribution and mean return times, compute the expected number of steps to return to state 1 starting from state 1.概率简单数值题未尝试免费487Mean Return Time on a Doubly Stochastic ChainA Markov chain on \ 0, 1, 2, 3\ has a doubly stochastic transition matrix (every row and every column sums to 1). Without knowing the specific entries of P, determine the expected number of steps to return to state 0 starting from state 0. Justify your answer.概率简单数值题未尝试免费491Hitting Time with State-Dependent Self-LoopsA Markov chain on \ 0, 1, 2, 3\ has transition probabilities: from state i (0 \le i \le 2), the chain stays at i with probability i 4 , moves to i+1 with probability 1 - i 4 . State 3 is absorbing. Compute E[T 3 \mid X 0 = 0].概率简单数值题未尝试免费497Hitting Time with Harmonically Increasing Forward ProbabilityA Markov chain on \ 0, 1, 2, 3\ has transitions: from state i (0 \le i \le 2), move to i+1 with probability i+1 i+2 and stay at i with probability 1 i+2 . State 3 is absorbing. Compute E[T 3 \mid X 0 = 0].概率简单derivation未尝试免费501Biased Gambler's Ruin ProbabilityA gambler starts with \3 and plays a sequence of independent rounds. Each round she wins \1 with probability p = 0.4 and loses \1 with probability q = 0.6. She stops when her fortune reaches \0 (ruin) or \8. What is the probability that she is ruined?概率简单数值题未尝试免费527Hitting Time to the Antipodal Vertex on a CycleA simple random walk moves on the cycle graph C 8 (vertices 0, 1, \ldots, 7 arranged in a circle). At each step, the walker moves clockwise or counterclockwise with equal probability \tfrac 1 2 . Starting at vertex 0, what is the expected number of steps to reach the antipodal vertex 4 for the first time?概率简单数值题未尝试免费531Cover Time of the Path Graph P₃A random walk moves on the path graph P 3 with vertices \ 1, 2, 3\ and edges \ 1 - 2, 2 - 3\ . At each step, the walker moves to a uniformly random neighbor (so from vertex 2 it goes to 1 or 3 each with probability \tfrac 1 2 , and from vertex 1 or 3 it moves deterministically to 2). Starting at vertex 1, what is the expected number of steps to visit all three vertices (the cover time)?概率简单数值题未尝试免费536Expected Return Time on the Complete Graph K₄A simple random walk moves on the complete graph K 4 (four vertices, every pair connected). At each step, the walker moves to one of the 3 neighbors chosen uniformly at random. Starting at a vertex v, what is the expected number of steps to return to v for the first time?概率简单数值题未尝试免费537Hitting Time Between Leaves on the Star Graph S₅The star graph S 5 has a central hub vertex c connected to 4 leaf vertices \ \ell 1, \ell 2, \ell 3, \ell 4\ . A simple random walk at each step moves to a uniformly random neighbor: from the hub, it goes to each leaf with probability \tfrac 1 4 ; from any leaf, it goes to c with probability 1. Starting at leaf \ell 1, what is the expected number of steps to reach leaf \ell 2 for the first time?概率简单数值题未尝试免费546Hitting Time on the Path Graph P₅A simple random walk moves on the path graph P 5 with vertices \ 0,1,2,3,4\ and edges connecting consecutive vertices. At the interior vertices (1, 2, 3), the walker moves left or right with equal probability \tfrac 1 2 . At the endpoints (0 and 4), the walker moves to the unique neighbor with probability 1. Starting at vertex 0, what is the expected number of steps to reach vertex 4 for the first time?概率简单数值题未尝试免费