5. If Xi, X2... X20 iid N(2,10) and Y 2(X - 4), find the mean and...
2. (10pts) Let X1, X2, , X20 be an i.i.d. sannple from a Normal distribution with mean μ and variance σ2, ie., Xi, X2, . . . , X20 ~ N(μ, σ2), with the density function Also let 20 20 10 20 -20 19 i-1 ー1 (a) (5pts) What are the distributions of Xi - X2 and (X1 - X2)2 respectively? Why? (b) (5pts) what are the distributions of Y20( and 201 ? Why? (X-μ)2 2. (10pts) Let X1, X2,...
Exercise 3.11 Suppose XI , X2. . . . , Xn are iid with mean 0 and variance 1 Show that Σ-M 0. Vnlog n
Suppose that Xi are IID normal random variables with mean 2 and variance 1, for i = 1, 2, ..., n. (a) Calculate P(X1 < 2.6), i.e., the probability that the first value collected is less than 2.6. (b) Suppose we collect a sample of size 2, X1 and X2. What is the probability that their sample mean is greater than 3? (c) Again, suppose we collect two samples (n=2), X1 and X2. What is the probability that their sum...
Let Xi, , X. .., Exp(β) be IID. Let Y max(Xi, , h} Find the probability density function of Y. İlint: Y < y if and only if XS for i 1,,n.
1 [3]. Let X1,X2, X3 be iid random variables with the common mean --1 2-4 and variance σ Find (a) E (2X1 - 3X2 + 4X3); (b) Var(2X1 -4X2); (c) Cov(Xi - X2, X1 +2X2).
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
8. Let Xi be iid N(μ, σ2) random variables. Define Y-Σ, Xi-Find the distribution of Y. a.
. Xi , X2, . . . , xn iid N(μ, σ2). Assume μ is known; show that θ Exactly 6.4-2 A)2 is the MLE for σ2 and show that it is unbiased. -ni(x-
Suppose that Xi,X2, , Х,, is an iid exponential (0) sample, where E(X) is unknown, and define Y, -X?, for i 1,2,.., n (a) Use the CLT to derive large-sample distribution of a properly centered and scaled version of (X, Y). (b) Find a consistent estimator of the covariance matrix in part (a). For the most part, "con sistency" means "convergence in probability."