第 3 / 23 页
非代码面试题
显示 20 / 453 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
368Second Moment of a Scale-Mixed Normal via TowerLet \Theta be drawn uniformly from \ 1, 2, 3\ and, given \Theta = , let X \mid \Theta = \sim N(0, ). Using the tower property, compute E[X 2] and E[X 4].概率中等数值题未尝试免费371Beta-Uniform Prior on Binomial Success ProbabilityLet P \sim Uniform (0,1) and, given P = p, let X \mid P = p \sim Binomial (10, p). Using the tower property, find E[X].概率简单数值题未尝试免费372Expected Maximum of Correlated Bernoullis via Indicator and TowerLet U \sim Uniform (0,1) and, given U, let X and Y be conditionally i.i.d.\ Bernoulli (U). Define M = \max(X, Y). Using the tower property and the indicator representation M = 1 \ X \ge 1 or Y \ge 1\ , find E[M].概率简单数值题未尝试免费373Two-Step Tower in an Additive Bernoulli Markov ChainLet X 1 \sim Uniform \ 0, 1\ . Given X 1, let X 2 = X 1 + B 1 where B 1 \sim Bernoulli (1/2) independent of X 1. Given X 2, let X 3 = X 2 + B 2 where B 2 \sim Bernoulli (1/2) independent of everything else. (a) Using the tower property E[X 3 \mid X 1] = E[E[X 3 \mid X 2] \mid X 1], find E[X 3 \mid X 1] and E[X 3]. (b) Using Eve's law, find Var (X 3).概率中等数值题未尝试免费376Distribution of the Cube of a Uniform Random VariableLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = X 3.概率简单derivation未尝试免费377Distribution of the Exponential of a Uniform via JacobianLet X \sim Uniform (0,1). Use the change-of-variables (Jacobian) formula to find the PDF of Y = e X.概率简单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381Negative Log of a Uniform Yields an ExponentialLet X \sim Uniform (0,1). Using the CDF method, derive the PDF of Y = -\ln X and identify the resulting distribution.概率简单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386Affine Transformation of a Normal Random VariableLet X \sim N( , 2) with a 0 and b \in R . Using the Jacobian formula, show that Y = aX + b is normally distributed and state its parameters.概率简单derivation未尝试免费388Odds Transformation of a Beta Variable Yields Beta PrimeLet X \sim Beta (a, b) with a, b > 0. Use the change-of-variables formula to derive the PDF of Y = \dfrac X 1 - X and identify the resulting distribution.概率中等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390Linear Transformation of a Multivariate Normal via MGFLet X \sim N(\boldsymbol , \boldsymbol \Sigma ) be a p-dimensional normal random vector, and let A be a fixed m p matrix. Using the moment-generating function, prove that Z = A X is multivariate normal and determine its mean vector and covariance matrix.概率困难derivation未尝试免费391Square Root of an Exponential Random VariableLet X \sim Exp (1). Use the change-of-variables formula to derive the PDF of Y = X and identify the resulting distribution.概率简单derivation未尝试免费392Tangent of a Uniform Variable Yields the Cauchy DistributionLet X \sim Uniform (- /2, /2). Derive the PDF of Y = \tan(X) using the change-of-variables formula and identify the resulting distribution.概率中等derivation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396Distribution of the Maximum of n Uniform Random VariablesLet X 1, \ldots, X n \sim iid Uniform (0,1). Derive the CDF and PDF of M = \max(X 1, \ldots, X n).概率简单derivation未尝试免费