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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