第 1 / 3 页
非代码面试题
显示 20 / 48 道匹配题目
答题状态：未尝试未正确已正确
ID题目领域难度题型进度权限
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287Robust Same-Rank Pairs in a Poker HandYou are dealt a 5-card hand uniformly at random from a standard 52-card deck. A same-rank pair is an unordered pair of cards in your hand that share the same rank (e.g., two Kings). Find the expected number of same-rank pairs in your hand. 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288Robust Isolated Vertices in a Random GraphIn the Erdos-Renyi random graph model G(n,p), each of the \binom n 2 possible edges among n labeled vertices is included independently with probability p. A vertex is isolated if it has no edges. Find the expected number of isolated vertices. 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296Robust Heads-Tails Transitions in Coin FlipsYou flip a fair coin n times independently. A transition at position i (for 1 \le i \le n-1) occurs when flip i and flip i+1 differ (one is heads and the other tails). Find the expected number of transitions. 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297Robust Distinct Birthdays in a GroupA group of n people each have a birthday chosen independently and uniformly at random from 365 days. A day is represented if at least one person in the group has that birthday. Find the expected number of represented days. 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333Robust Law of Total Variance with a Random Number of Coin FlipsYou first draw N \sim Poisson (5). Then, given N = n, you flip a fair coin n times and let X be the number of heads. What is Var (X)? 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334Robust Covariance of Overlapping Sums of Independent VariablesLet X, Y, Z be independent random variables with Var (X) = 1, Var (Y) = 2, and Var (Z) = 3. Define U = X + Y and V = Y + Z. Compute Cov (U, V) and Corr (U, V). 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