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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