第 1 / 4 页
非代码面试题
显示 20 / 64 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
150Red Dice Meet Blue DiceYou simultaneously roll 3 red fair six-sided dice and 3 blue fair six-sided dice. What is the probability that the sum of the red dice equals the sum of the blue dice?概率困难数值题未尝试免费200Full Distribution of Empty Urns via Stirling NumbersSix distinguishable balls are thrown independently and uniformly at random into 5 distinguishable urns. Let E denote the number of empty urns. Derive the probability mass function P(E = k) for every possible value of k, expressing each probability as an exact fraction.概率困难derivation未尝试免费212Geometric Moments via the Probability Generating FunctionLet X \sim Geometric (p) with P(X = k) = (1-p) k-1 p for k = 1, 2, 3, \ldots (a) Derive the probability generating function G X(s) = E[s X] in closed form for |s| < 1 1-p . (b) Using G X, compute E[X] and Var (X). Recall: E[X] = G X'(1) and Var (X) = G X''(1) + G X'(1) - [G X'(1)] 2.概率简单derivation未尝试免费215Distribution of Dice Sums via Probability Generating FunctionsLet X 1, X 2, \ldots, X n be iid rolls of a fair d-sided die, so each X i is uniform on \ 1, 2, \ldots, d\ . Let S n = X 1 + X 2 + \cdots + X n. (a) Derive the PGF G X 1 (s) = E[s X 1 ] in closed form. (b) Write the PGF of S n and use it to derive E[S n] and Var (S n). (c) For n = 3 fair six-sided dice (d = 6), use the PGF to find P(S 3 = 10). (d) Explain how the coefficient-extraction approach relates to the classical stars-and-bars counting with inclusion-exclusion for this problem.概率困难derivation未尝试免费228Moment Generating Function of the Gamma DistributionLet X \sim Gamma ( , ) with PDF f(x) = \Gamma( ) x -1 e - x for x > 0. Derive the moment generating function M X(t) = E[e tX ] for t < . Then use M X(t) to compute E[X] and Var (X).概率中等derivation未尝试免费230Beta Distribution: Derivation, Moments, and ConnectionsConsider the Beta distribution with PDF f(x) = \Gamma( + ) \Gamma( )\Gamma( ) x -1 (1-x) -1 for x \in (0,1). (a) Verify that f integrates to 1 by showing \int 0 1 x -1 (1-x) -1 \,dx = \Gamma( )\Gamma( ) \Gamma( + ) using the integral representation of the Gamma function. (b) Derive E[X] = + and Var (X) = ( + ) 2( + +1) . (c) Show that Beta (1,1) reduces to Uniform (0,1). (d) If Y 1 \sim Gamma ( , 1) and Y 2 \sim Gamma ( , 1) are independent, explain (without full proof) why Y 1 Y 1+Y 2 \sim Beta ( , ).概率困难derivation未尝试免费262Waiting for ABAB in a Uniform ABC StreamEach symbol in an iid stream is chosen uniformly from \ A,B,C\ . Find the expected waiting time until ABAB first appears.概率困难derivation未尝试免费264Expected Time Until ABC or CBAA symbol stream is iid and uniform on \ A,B,C\ . What is the expected number of symbols until either ABC or CBA first appears?概率简单数值题未尝试免费267Nonuniform Stream Waiting for ABACAn iid source emits A,B,C with probabilities rac 1 2 , rac 1 3 , rac 1 6 respectively. Find the expected waiting time until ABAC first appears.概率困难derivation未尝试免费268ABCA in a Four-Symbol StreamEach symbol of an iid stream is chosen uniformly from \ A,B,C,D\ . Compute the expected waiting time until ABCA first appears.概率困难derivation未尝试免费269Finish ABCA After Current Suffix ABCAn iid stream over \ A,B,C,D\ is uniform. Suppose the current observed suffix is exactly ABC. From this point onward, what is the expected additional number of symbols until ABCA first appears?概率简单数值题未尝试免费273Return-and-Overlap Pattern 1-2-1 on a DieA fair six-sided die is rolled repeatedly. Find the expected waiting time until the consecutive pattern 1,2,1 first appears.概率困难derivation未尝试免费274Either 123 or 321 on a Fair DieA fair six-sided die is rolled repeatedly. Let T be the first time that either 1,2,3 or 3,2,1 appears as a consecutive length-3 block. Compute E[T].概率困难derivation未尝试免费275Waiting for HHTHA fair coin is flipped repeatedly. What is the expected number of flips until HHTH first appears?概率中等derivation未尝试免费503Asymmetric Step-Size Gambler's RuinA gambler starts with \3. Each round she wins \1 with probability \tfrac 2 3 or loses \2 with probability \tfrac 1 3 . She stops when her fortune first hits \0 or below (ruin) or reaches \5 or above (victory). Note that if her fortune is \1 and she loses, she goes to -\1, which counts as ruin. What is the probability that she achieves victory?概率中等数值题未尝试免费505Gambler's Ruin with a Partially Reflecting BarrierConsider a Markov chain on \ 0, 1, 2, \ldots\ with absorbing state 0. From state k \ge 1, the chain moves to k+1 with probability p and to k-1 with probability q = 1 - p, where 0 < p < 1. However, there is a **reflecting barrier** at state N: from state N, the chain moves to N-1 with probability 1 (it is always pushed back). Starting from state k with 1 \le k \le N: **(a)** Find the probability r k of eventual absorption at 0. **(b)** For p = q = \tfrac 1 2 and N = 4, compute r 2 and the expected absorption time E[T \mid X 0 = 2].概率困难derivation未尝试免费512Asymmetric Jump Ruin with Upward LeapA gambler on \ 0, 1, \ldots, 6\ starts at position 3. Each round, she jumps +2 with probability 1/3 and -1 with probability 2/3. Positions 0 and 6 are absorbing (but a jump from 5 that would land at 7 is truncated to 6, i.e., from state 5 the gambler moves to 6 with probability 1/3 and to 4 with probability 2/3). What is the probability that the gambler is absorbed at 0?概率中等数值题未尝试免费519Ruin Probability via Probability Generating FunctionsConsider a random walk on \ 0, 1, 2, \ldots, N\ with absorbing barriers at 0 and N. At each step, the walker moves +1 with probability p and -1 with probability q = 1 - p. Let G k(s) = E[s \tau \mid X 0 = k] be the probability generating function of the absorption time \tau, for 0 < k < N. (a) Derive a recurrence relation for G k(s). (b) For N = 4, k = 2, and p = q = 1/2, find G 2(s) explicitly and use it to compute E[\tau] and Var (\tau).概率困难derivation未尝试面试订阅2721Unit and Domino Tilings of a Nine-Cell StripA strip of length 9 is tiled using monomers of length 1 and dominoes of length 2. Derive the generating function for the number a n of tilings, and compute a 9.脑筋急转弯简单数值题未尝试面试订阅2722Tiling With 1-, 2-, and 4-Step BlocksLet a n be the number of ways to tile a strip of length n using blocks of lengths 1, 2, and 4. Write the generating function A(x) and compute a 10.脑筋急转弯中等derivation未尝试面试订阅