Let X1, X2, . . . , Xn be a random sample from N(0, σ2), where n is odd. Let Y and Z be the mean and median of the sample. Argue that Y and Z − Y are independent so that the variance of Z is Var(Y) + Var(Z− Y). We know that Var(Y) = σ2/n, so that we could estimate the Var(Z − Y) by Monte Carlo. This might be more efficient than estimating Var(Z) directly since Var(Z − Y) ≤ Var(Z). This scheme is often called the Monte Carlo Swindle.
Answer to relevant QuestionsLet 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, ...Consider a random sample X1, X2, . . . , Xn from a distribution with pdf Let θ have a prior pdf that is gamma with α = 4 and the usual θ = 1/4. Find the conditional mean of θ, given that X1 = x1, X2 = x2, . . . , Xn = ...In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same ...Let X and Y equal the hardness of the hot and cold water, respectively, in a campus building. Hardness is measured in terms of the calcium ion concentration (in ppm). The following data were collected (n = 12 observations of ...Let p equal the proportion of Americans who favor the death penalty. If a random sample of n = 1234 Americans yielded y = 864 who favored the death penalty, find an approximate 95% confidence interval for p.
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