ULLL Dsu i8 an unbiased estimate of Umin 4. The random variables X1, X2, . ....
. If X1, X2,..., Xn are independent random variables with common mean μ and variances σ1, σ2, . . ., σα , prove that Σί (Xi-T)2/[n(n-1)] is an ว. 102n unbiased estimate of var[X] 3. Suppose that in Exercise 2 the variances are known. LeTw Σί uiXi
3. Let X1, X2, . . . , Xn be random variables with a common mean μ. Sup- pose that cov[Xi, xj] = 0 for all i and A such that j > i+1. If 仁1 and 6 VECTORS OF RANDOM VARIABLES prove that = var X n(n- 3)
Please solve these questions 1. Suppose that X1, X2, and Xs are random variables with common mean μ and variance matrix Find E(X1 +2X1X2-4X2X3 + X ]. 2. If X1, X2,..., X, are independent random variables with common mean (n - 1)] is an μ and variances σ?, σ2, .. ., σ unbiased estimate of varf , prove that Σ,(X,-X)2/[n 3. Suppose that in Exercise 2 the variances are known. Let X,-Σ,wa, be an unbiased estimate of μ (i.e., Σί...
lue lalloni Variables, each with it variance, find VarfX]. 4. If X1,X2, ,An are random variables satisfying Xi+1 (is 1, 2, . . . ,n-1), where ρ is a constant, and var[X1-σ2, find Var(X)
5. Let X1, X2, . . . , Xn be independently distributed as N(u, σ2). Define 71 -1 /Users/rumi3/Downloads/Linear-Regression-Analysis-Seber pdf MOMENT GENERATING FUNCTIONS AND INDEPENDENCE 13 and (a) Prove that var[S2-2σ4/(n-1). (b) Show that Q is an unbiased estimate of σ2. (c) Find the variance of Qand hence show that as n → oo, the effi- ciency of Q relative to S2 is
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
5. Let X1, X2, . .. , Xn be independently distributed as N(μ, σ2). Define 7t n-1 ー1 Users/rumi3/Downloads/Linear-Regression-Analysis-Seber.pdf MOMENT GENERATING FUNCTIONS AND INDEPENDENCE 13 and n-1 2(n-1 i=1 (a) Prove that var[S2-2c4/(n-1). b) Show that Q is an unbiased estimate ofa (c) Find the variance of Q and hence show that as n → oo, the effi- ciency of Q relative to S2 is
3. Suppose that X1, X2, , Xn are independent random variables with the same expectation μ and the same variance σ2. Let X--ΣΑι Xi. Find the expectation and variance of
How do you show this? 1.2.12. Accept the following definition. Discrete random variables X1, X2,.. , Xn, taking values in Ai, A2,..., An, are said to be independent if (1) P(Xi = ai , . . . ,x, = an) =11P(X, = a.) 仁1 for all ai E A1,., an E An. Then prove that random variables in any subsequence of a finite sequence of independent random variables are independent.
Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we have two estimators 1 (1) Are both estimators unbiased estimatros for θ? (2) Which is a better estimator?