Let X1, X2, . . . , Xn be a random sample from a Poisson distribution with mean λ > 0. Find the conditional probability P(X1 = x1, . . . , Xn = xn | Y = y), where Y = X1 + · · · + Xn and the nonnegative integers x1, x2, . . . , xn sum to y. Note that this probability does not depend on λ.
Answer to relevant QuestionsLet X1, X2, . . . , Xn be a random sample from a distribution with pdf f(x; θ) = θxθ−1, 0 < x < 1, where 0 < θ. (a) Find a sufficient statistic Y for θ. Let Y be the largest order statistic of a random sample of size n from a distribution with pdf f(x | θ) = 1/θ, 0 < x < θ. Say θ has the prior pdf where α > 0, β > 0. (a) If w(Y) is the Bayes estimator of θ and [θ − ...Let X1, X2, ... , Xn be a random sample of size n from the normal distribution N(μ, σ2). Calculate the expected length of a 95% confidence interval for μ, assuming that n = 5 and the variance is (a) Known. (b) Unknown. Let S2X and S2Y be the respective variances of two independent random samples of sizes n and m from N(μX, σ2X) and N(μY, σ2Y). Use the fact that F = [S2Y/σ2Y]/[S2X/σ2X] has an F distribution, with parameters r1 = m− ...Let p equal the proportion of Americans who select jogging as one of their recreational activities. If 1497 out of a random sample of 5757 selected jogging, find an approximate 98% confidence interval for p.
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