8. Let Xi be iid N(μ, σ2) random variables. Define Y-Σ, Xi-Find the distribution of Y. a.
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
Given a continuous random variable, prove that s--a:G-x) 2 converges to σ2 as Σ-1(xi-x) 2 converges to σ2 as n-1
Given a continuous random variable, prove that s--a:G-x) 2 converges to σ2 as Σ-1(xi-x) 2 converges to σ2 as n-1
4. Let X1,X2, x 2) distribution, and let sr_ Ση:1 (Xi-X)2 and S2 n-l Σηι (Xi-X)2 be the estimators of σ2. (i) Show that the MSE of S" is smaller than the MSE of S2 (ii) Find ElvS2] and suggest an unbiased estimator of σ. n be a random sample from N (μ, σ
For i 1, let X G1/2 be distributed Geometrically with parameter 1/2 Define Xi -2 Approximate P (-1 < -2) with large enough n.
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
4. Suppose Σί, 1 Xi-1, ΣΙ-, x-2, and n-5. Evaluate the followings: a) Σί-1[Xi + X) b) 5. Let X-1 Σ., x1-1 with n-100. (i) Obtain ΣΙ.i Xu (ii) Evaluate Σ.1x,-X) and X +1
Suppose that Xi, X2,..., Xn is an iid sample from 20 for x R and σ 〉 0. (a) Derive a size α likelihood ratio test (LRT) of H0 : σ (b) Derive the power function β(o) of the LRT 1 versus H1 : σ 1.
Let xi, i 1, 2, 3, , be a sequence of nonnegative numbers such that Σ x.-1 and consider the random variable X whose probability function is defined by: x, for x=x1, x2, X3, 0, for all other x What is the variance of X? i= 1