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- Let X be a random variable with mean μ and variance _2. Show that E[(X − b)2], as a function of b, is minimized when b = μ.If X is a uniformly distributed random varibale with a=8 and b=13, then Calculate the mean and variance of X? Round to three decimal placesLet X1, X2, ... , Xn be a random sample, normally distributed with mean μ and variance σ2If σ2 is unknown, find a minimum value for n to guarantee, with probability 0.90, that a 0.95 CI for μ will have length no more than σ/4
- Let X1, X2,...Xn be a random sample of size n from a normal distribution with mean u and variance o2. Let Xn denote the sample average, defined in the usual way. PROVE E [Xn] = uLet 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.Let (X1,...,Xn) be a random sample of random variables with EX21 < ∞. Consider estimating µ = EX1 under the squared error loss. Show that (i) any estimator of the form a ¯ X+bis inadmissible, where ¯ X is the sample mean, a and b are constants, and a>1;
- Let X1,...,Xn be a random sample from a distribution with mean μ and variance σ2. Find the value of the following quantities (as a function of μ and σ2). (d) Var(4X1 −3X2 +2X3 −X4) (e) E{(X1 − X2)2} (f) {E(X1 − X2)}2If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal places