Let X1, X2, ..., Xn ~ exp(A) be a random sample. Show that X is sufficient for 1/A.
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- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.