第 1 / 2 页
非代码面试题
显示 20 / 25 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
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387Probability Integral Transform (Inverse CDF Method)Let F X be a continuous, strictly increasing CDF and U \sim Uniform (0,1). Prove that Y = F X -1 (U) has CDF F X. Conversely, show that if X has CDF F X, then F X(X) \sim Uniform (0,1).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