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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302Robust Matching Colors in a LineThere are n people standing in a line. Each person independently and uniformly picks one of three colors: red, green, or blue. What is the expected number of adjacent pairs (i, i+1) who chose the same color? 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303Robust Coincident RollsYou roll a fair k-sided die n times independently. Let M be the number of pairs (i, j) with 1 \le i < j \le n such that roll i equals roll j. Find E[M]. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation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315Robust Singleton Coupons After Random DrawsA collector draws m coupons independently and uniformly at random from n types. A coupon type is called a singleton if it appears exactly once among the m draws. Find the expected number of singleton types. 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322Robust Adjacent Matches in Dice RollsRoll a fair six-sided die n times independently, producing a sequence D 1, D 2, \dots, D n. An adjacent match occurs at position i (for 1 \le i \le n - 1) if D i = D i+1 . Find the expected number of adjacent matches. 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323Robust Overlap of Two Random SubsetsLet S and T be two subsets of \ 1, 2, \dots, n\ , each chosen independently and uniformly at random from all \binom n k subsets of size k (where 1 \le k \le n). Find the expected size of their intersection |S \cap T|. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费324Robust Monochromatic Edges in a Random ColoringEach vertex of the complete graph K n is independently colored red or blue, each with probability \tfrac 1 2 . An edge is monochromatic if both its endpoints have the same color. Find the expected number of monochromatic edges. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费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