第 11 / 209 页
非代码面试题
显示 20 / 4169 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
232Sum of Two Independent Exponentials via ConvolutionLet X 1, X 2 be independent, each distributed Exponential ( ). Using the convolution formula, derive the PDF of Y = X 1 + X 2 and identify the resulting distribution.概率中等derivation未尝试免费233Mean and Variance of the Lognormal DistributionLet X be a lognormal random variable, meaning \ln X \sim N( , 2). Using the moment generating function of the normal distribution, derive E[X] and Var (X).概率中等derivation未尝试免费235Chi-Squared Distribution from First PrinciplesLet Z 1, \ldots, Z n be i.i.d. N(0,1) random variables and define Q = \sum i=1 n Z i 2. (a) Derive the PDF of Z 1 2 using the change-of-variables technique for transformations of continuous random variables. (b) Show that Z 1 2 \sim Gamma ( 1 2 , 1 2 ) by matching the PDF from (a) to the Gamma family. (c) Using the fact that the sum of independent Gamma random variables with the same rate parameter is Gamma, state the distribution of Q. (d) Derive E[Q] and Var (Q).概率困难derivation未尝试免费236Median of the Weibull DistributionThe Weibull distribution with shape k > 0 and scale > 0 has CDF F(x) = 1 - e -(x/ ) k , \quad x 0. (a) Derive a closed-form expression for the median of this distribution. (b) Compute the median when k = 2 and = 3.概率简单数值题未尝试免费237Mean and Variance of the Gamma Distribution via IntegrationLet X \sim Gamma ( , ) with PDF f(x) = \Gamma( ) x - 1 e - x for x > 0. (a) Derive E[X] by direct integration, using the fact that \int 0 \Gamma( ) x - 1 e - x \,dx = 1 for any > 0. (b) Derive E[X 2] similarly and compute Var (X).概率简单derivation未尝试免费238Non-Existence of the Mean for the Cauchy DistributionThe standard Cauchy distribution has PDF f(x) = 1 (1 + x 2) for x \in (- , ). (a) Show that E[|X|] does not exist by proving the integral \int 0 x (1+x 2) \,dx diverges. (b) What does this imply about the applicability of the Law of Large Numbers to the sample mean X n = 1 n \sum i=1 n X i when X i are i.i.d. Cauchy?概率中等derivation未尝试免费239Ratio of Independent Standard Normals is CauchyLet Z 1, Z 2 be independent N(0,1) random variables. Define R = Z 1 / Z 2. (a) Write the joint PDF of (Z 1, Z 2) and use the transformation (Z 1, Z 2) \mapsto (R, Z 2) = (Z 1/Z 2,\, Z 2) to derive the joint PDF of (R, Z 2). (b) Integrate out Z 2 to obtain the marginal PDF of R. (c) Identify the distribution of R.概率困难derivation未尝试免费240Beta Distribution and the Beta Function from Independent GammasLet X \sim Gamma ( , 1) and Y \sim Gamma ( , 1) be independent. (a) Define U = X X+Y and V = X + Y. Compute the Jacobian of the transformation (X, Y) \mapsto (U, V). (b) Derive the joint PDF of (U, V) and show that U and V are independent. (c) Identify the marginal distribution of U and prove that B( , ) = \Gamma( )\,\Gamma( ) \Gamma( + ) , where B( , ) = \int 0 1 u -1 (1-u) -1 \,du.概率困难derivation未尝试免费241Tail Probability and Conditional Expectation of the Pareto DistributionThe Pareto distribution with shape > 0 and scale x m > 0 has PDF f(x) = \, x m x +1 , \quad x x m. (a) Derive P(X > t) for t x m. (b) Derive E[X \mid X > t] for > 1 and t x m. (c) Compute both quantities for = 3, x m = 1, t = 2.概率简单数值题未尝试免费242Distribution of the Sum of Two Independent UniformsLet X and Y be independent Uniform (0, 1) random variables. Define S = X + Y. (a) Using the convolution formula f S(s) = \int - f X(t)\, f Y(s - t)\, dt, derive the PDF of S for all s \in R . (b) Sketch the PDF and identify the distribution by name. (c) Compute P(S > 1.5).概率中等derivation未尝试免费243Inverse Transform Sampling for the Exponential DistributionLet U \sim Uniform (0, 1). (a) For a continuous random variable X with strictly increasing CDF F, prove that F(X) \sim Uniform (0,1). (b) Use part (a) to argue that X = F -1 (U) has CDF F. (c) For X \sim Exp ( ) with CDF F(x) = 1 - e - x (x 0), derive F -1 explicitly and write a one-line formula for generating exponential samples from uniform samples.概率中等derivation未尝试免费244Distribution of the Square of a Standard Normal via Change of VariablesLet X \sim N(0, 1). (a) The transformation Y = X 2 is not monotone. Using the CDF method, derive f Y(y) by first computing F Y(y) = P(X 2 y) and then differentiating. (b) Alternatively, apply the non-monotone change-of-variables formula: if Y = g(X) and g has exactly two branches x 1(y) and x 2(y) satisfying g(x i) = y, then f Y(y) = \sum i=1 2 f X(x i(y))\, | dx i dy |. Verify you get the same answer. (c) Identify the resulting distribution and express it as a Gamma distribution.概率困难derivation未尝试免费245Maximum Entropy Property of the Normal DistributionThe differential entropy of a continuous random variable X with PDF f is h(X) = -\int - f(x) \ln f(x)\, dx. (a) Among all continuous distributions on R with mean and variance 2, use the method of Lagrange multipliers to show that the PDF maximizing h(X) satisfies \ln f(x) = -1 + \lambda 0 + \lambda 1 x + \lambda 2 x 2 for some constants \lambda 0, \lambda 1, \lambda 2. (b) Determine \lambda 0, \lambda 1, \lambda 2 by enforcing the three constraints (\int f = 1, \int xf = , \int x 2 f = 2 + 2) and show that f is the N( , 2) PDF. (c) Compute h(X) for X \sim N( , 2).概率困难derivation未尝试免费246Rayleigh Distribution from Independent NormalsLet X and Y be independent N(0, 2) random variables. Define R = X 2 + Y 2 . (a) Using the CDF method, derive the PDF of R for r 0. (b) Identify the distribution by name and compute E[R]. (c) Evaluate E[R] and Var (R) numerically for = 1.概率简单derivation未尝试免费247Erlang Distribution as Poisson Process Waiting TimeArrivals follow a Poisson process with rate > 0. Let T k denote the time of the k-th arrival. (a) Using the relationship P(T k > t) = P(N(t) < k) where N(t) \sim Poisson ( t), write the CDF of T k and differentiate to obtain its PDF. (b) Identify the distribution by name and state its mean and variance. (c) For = 1 and k = 3, compute P(T 3 > 2) as an exact numerical value.概率简单数值题未尝试免费248Mean, Variance, and Bimodality of a Gaussian MixtureA random variable X is drawn from a mixture of two normals: with probability p we sample from N(\mu 1, \sigma 1 2) and with probability 1-p from N(\mu 2, \sigma 2 2). (a) Derive E[X] and Var (X) in terms of p, \mu 1, \mu 2, \sigma 1 2, \sigma 2 2. (b) For the symmetric case p = 1/2 and \sigma 1 = \sigma 2 = , show that the mixture PDF is bimodal if and only if |\mu 1 - \mu 2| > 2 . (c) Compute E[X] and Var (X) for p = 1/2, \mu 1 = -2, \mu 2 = 2, \sigma 1 = \sigma 2 = 1, and verify that this mixture is bimodal.概率中等derivation未尝试免费249Order Statistics of the Uniform Distribution: Min, Max, and RangeLet X 1, X 2, \ldots, X n be iid Uniform (0, 1) random variables. Denote the minimum by X (1) and the maximum by X (n) . (a) Derive the PDF of X (1) and the PDF of X (n) . (b) Compute E[X (1) ] and E[X (n) ]. (c) The range is W = X (n) - X (1) . Compute E[W] for n = 5.概率中等数值题未尝试免费250The Folded Normal Distribution: PDF and Moments of |X|Let X \sim N( , 2) with 0. Define Y = |X|. (a) Derive the PDF of Y for y 0 by writing P(Y y) = P(-y X y) and differentiating. (b) Show that when = 0 the PDF simplifies to the half-normal distribution f Y(y) = 2 \, e -y 2/(2 2) for y 0. (c) Derive E[Y] and Var (Y) for general , . Express your answer using the standard normal PDF \phi and CDF \Phi. (d) Compute E[Y] and Var (Y) numerically for = 1, = 1.概率困难derivation未尝试免费251Remaining Time to HTH After Seeing HTA fair coin has been flipped for a while, and the current suffix of the observed sequence is exactly HT (that is, the last two flips are HT). From this point onward, what is the expected additional number of flips until HTH appears for the first time?概率简单数值题未尝试免费252Remaining Time to HTH After Seeing HA fair coin is being flipped repeatedly. You are told that the current observed suffix is exactly H. Starting from that suffix-state, what is the expected additional number of flips until HTH first appears?概率简单数值题未尝试免费