第 2 / 13 页
非代码面试题
显示 20 / 244 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
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359Tower Property with a Continuous Mixing ParameterLet U \sim Uniform (0,1) and, given U = u, let X \sim Geometric (u) (number of trials until first success, so P(X = k \mid U = u) = (1-u) k-1 u for k = 1, 2, \ldots). Find E[X] using the tower property.概率中等数值题未尝试免费378Distribution of the Sum of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the convolution formula, derive the PDF of Z = X + Y.概率中等derivation未尝试免费379Distribution and Mean of the Maximum of Two ExponentialsLet X and Y be independent Exp (1) random variables. Define M = \max(X, Y). (a) Derive the PDF of M. (b) Compute E[M].概率中等数值题未尝试免费380Distribution of the Ratio of Two Independent ExponentialsLet X and Y be independent Exp (1) random variables. Define R = X/Y. (a) Using the transformation (R, S) = (X/Y,\, Y), compute the joint density f R,S via the Jacobian and then marginalize over S to find the PDF of R. (b) Identify f R as a named distribution and verify by computing P(R \le 1) using a symmetry argument.概率困难derivation未尝试免费382Square of a Standard Normal Is Chi-Squared(1)Let X \sim N(0,1). Using the CDF method, derive the PDF of Y = X 2 and identify the resulting distribution.概率中等derivation未尝试免费383Reciprocal Transform of an Exponential VariableLet X \sim Exp (1). Define Y = \dfrac 1 1 + X . (a) Use the change-of-variables formula to derive the PDF of Y. (b) Compute E[Y].概率中等derivation未尝试免费384Distribution of the Product of Two Independent UniformsLet X and Y be independent Uniform (0,1) random variables. Using the transformation (W, V) = (XY,\, Y), derive the PDF of W = XY.概率中等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389Ratio of Independent Gammas Yields a Beta DistributionLet X \sim Gamma ( , 1) and Y \sim Gamma ( , 1) be independent. Using the transformation (W, S) = \bigl(X/(X+Y),\; X+Y\bigr): (a) Compute the Jacobian of the inverse map. (b) Derive the joint density f W,S and marginalize to show W \sim Beta ( , ). (c) Show that W and S are independent.概率困难derivation未尝试免费393Sum of Squares of Two Standard NormalsLet X 1, X 2 \sim iid N(0,1). Define R = X 1 2 + X 2 2. (a) Switching to polar coordinates (X 1, X 2) = (r\cos , r\sin ), derive the joint density of (R, \Theta) where R = X 1 2 + X 2 2 and \Theta = \arctan(X 2/X 1). (b) Marginalize over \Theta to find the PDF of R and identify its distribution.概率中等multi part未尝试免费394Ratio of Independent Standard Normals Is CauchyLet X 1, X 2 \sim iid N(0,1). Using the transformation (Y, V) = (X 1/X 2,\, X 2): (a) Derive the joint density f Y,V (y,v). (b) Integrate out V to obtain the marginal PDF of Y = X 1/X 2 and identify the distribution.概率困难derivation未尝试免费395Exponential of a Normal: The Log-Normal DistributionLet X \sim N( , 2) and define Y = e X. (a) Derive the PDF of Y using the change-of-variables formula. (b) Using the MGF of the normal distribution, compute E[Y] and Var (Y). (c) Show that the median of Y is e and explain why E[Y] > median (Y) when > 0.概率困难multi part未尝试免费400Deriving the Fisher F-Distribution from Chi-Squared VariablesLet X \sim \chi 2(m) and Y \sim \chi 2(n) be independent. Define F = X/m Y/n . (a) Using the transformation (F, W) = \bigl( nX mY ,\; Y\bigr), compute the Jacobian and derive the joint density f F,W . (b) Integrate out W to obtain the marginal PDF of F and verify it matches the F(m, n) distribution. (c) Show that E[F] = \dfrac n n-2 for n > 2.概率困难multi part未尝试免费402Distribution of the Minimum of Exponential Random VariablesLet X 1, \ldots, X n be independent Exp ( ) random variables. Derive the distribution of X (1) = \min(X 1, \ldots, X n).概率简单derivation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