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189Probability of a Heavily Loaded UrnSix distinguishable balls are thrown independently and uniformly at random into 4 distinguishable urns. What is the probability that at least one urn contains 3 or more balls? Give an exact fraction.概率困难数值题未尝试免费205Hypergeometric Mean and Variance via Indicator VariablesAn urn contains 20 balls: 8 red and 12 blue. You draw 5 balls without replacement. Let X be the number of red balls drawn. Define indicator variables X i = 1 \ ball i is red \ for each draw i = 1, \ldots, 5. (a) Use linearity of expectation to find E[X]. (b) Compute Cov (X i, X j) for i \ne j and use it to derive Var (X). (c) Verify that your variance formula reduces to the binomial variance np(1-p) when N with K/N p held fixed.概率困难derivation未尝试免费229Standard Normal Tail and Interval ProbabilitiesLet Z \sim N(0,1). Using \Phi(1) \approx 0.8413 and \Phi(2) \approx 0.9772, compute: (a) P(|Z| > 2) and (b) P(1 < Z < 2). Explain each step using symmetry of the normal distribution.概率中等数值题未尝试免费285Robust Monochromatic Cliques in a Random Edge-ColoringEach edge of the complete graph K n is independently colored red or blue with equal probability 1 2 . For a fixed integer k \ge 2, find the expected number of monochromatic k-cliques (complete subgraphs on k vertices whose edges are all the same color). Express your answer in terms of n and k. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费286Robust Ascents in a Random PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . An ascent at position i (for 1 \le i \le n-1) is a position where (i) < (i+1). Find the expected number of ascents. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单derivation未尝试免费295Robust Cycles in a Random PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . Find the expected number of cycles in the cycle decomposition of . Express your answer as a familiar function of n. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费300Robust Common Edges of Two Random Spanning TreesLet T 1 and T 2 be two independent uniformly random spanning trees of the complete graph K n (each drawn uniformly at random from all n n-2 labeled spanning trees, independently of the other). Find the expected number of edges that belong to both T 1 and T 2. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费301Robust Descents in a Random PermutationLet be a permutation of \ 1, 2, \dots, n\ chosen uniformly at random. A descent is a position i \in \ 1, \dots, n-1\ where (i) > (i+1). What is the expected number of descents? Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费305Robust Unique Choices and Unique NeighborsThere are n people in a line, and each independently and uniformly picks an integer from \ 1, 2, \dots, k\ . A person is called unique if no other person picked the same number. (a) Using indicator variables, find E[U], the expected number of unique people. (b) A unique neighbor pair is a pair of adjacent people (i, i+1) who are both unique. Find E[N], the expected number of unique neighbor pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费306Robust Intermediate Positions in a PermutationLet be a uniformly random permutation of \ 1, 2, \dots, n\ . Call position i \in \ 2, \dots, n-1\ an intermediate position if (i) is strictly between (i-1) and (i+1), i.e., \min( (i-1), (i+1)) < (i) < \max( (i-1), (i+1)). What is the expected number of intermediate positions? Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费325Robust Comparable Pairs in Random PointsLet X 1, X 2, \dots, X n be independent and uniformly distributed on [0,1] d (the d-dimensional unit hypercube). Two points X i and X j are called comparable if one dominates the other coordinatewise, i.e., either X i \le X j in every coordinate or X j \le X i in every coordinate. Find the expected number of comparable pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费385Box-Muller Transform: From Uniforms to Independent NormalsLet U 1, U 2 be independent Uniform (0,1) random variables. Define Z 1 = -2\ln U 1 \,\cos(2 U 2), \qquad Z 2 = -2\ln U 1 \,\sin(2 U 2). (a) Compute the Jacobian of the inverse transformation from (Z 1, Z 2) back to (U 1, U 2). (b) Show that Z 1 and Z 2 are independent N(0,1) random variables.概率困难derivation未尝试免费390Linear Transformation of a Multivariate Normal via MGFLet X \sim N(\boldsymbol , \boldsymbol \Sigma ) be a p-dimensional normal random vector, and let A be a fixed m p matrix. Using the moment-generating function, prove that Z = A X is multivariate normal and determine its mean vector and covariance matrix.概率困难derivation未尝试免费398Additivity of Chi-Squared Distributions via MGFLet X \sim \chi 2(m) and Y \sim \chi 2(n) be independent. Using moment-generating functions, prove that X + Y \sim \chi 2(m + n).概率中等derivation未尝试免费399Absolute Value of a Standard Normal: The Half-Normal DistributionLet X \sim N(0,1) and define Y = |X|. (a) Derive the PDF of Y using the CDF method (note that Y = |X| is not monotone). (b) Compute E[Y] and Var (Y).概率中等multi part未尝试免费404Expected Range of Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). The range is defined as R = X (n) - X (1) . Derive a closed-form expression for E[R] as a function of n.概率中等derivation未尝试免费409Expected Spacing Between Consecutive Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1) and let X (0) = 0, X (n+1) = 1. Show that E[X (k+1) - X (k) ] = 1 n+1 for every k = 0, 1, \ldots, n, and compute this value for n = 4.概率中等数值题未尝试免费410Joint Density and Covariance of Two Uniform Order StatisticsLet X 1, \ldots, X n be iid Uniform (0,1). Consider the order statistics X (i) and X (j) with 1 \le i < j \le n.概率困难multi part未尝试面试订阅415Distribution of the Mid-Range for Uniform SamplesLet X 1, \ldots, X n be iid Uniform (0,1) with n \ge 2. The mid-range is defined as M = X (1) + X (n) 2 . Using the joint density of (X (1) , X (n) ), derive the PDF of M.概率困难derivation未尝试面试订阅418Expected Value of the Second Smallest ExponentialLet X 1, \ldots, X 5 be independent Exp (1) random variables. Derive E[X (2) ].概率中等derivation未尝试免费