. If X1, X2,..., Xn are independent random variables with common mean μ and variances σ1,...
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...
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)
ULLL Dsu i8 an unbiased estimate of Umin 4. The random variables X1, X2, . . . .xn have a common nonzero mean μ, a common variance σ2, and the correlation between any pair of random variables is ρ. (a) Find var (b) If and hence prove that-1/(n-1) バ1. 71 仁1 is an unbiased estimate of σ2, find a and b. Hence show that, i this case,
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
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.
Suppose that X1, X2 are two independent random variables with a common mean μ, but two different variances σ12 > σ22. Consider the family of estimators Wα = αX1 + (1−α)X2 where 0 ≤ α ≤ 1 (a) Show that Wα is an unbiased estimator of μ for any value of α. (b) Find the value of α which makes Wα as efficient as possible. Explain why the resulting formula makes sense.
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
Suppose that X1, ..., Xn is a random sample from a normal distribution with mean μ and variance σ2. Two unbiased estimators of σ2 are 1?n 1 i=1 σˆ12 =S2 = n−1 Find the efficiency of σˆ12 relative to σˆ2. (Xi −X̄)2, and σˆ2= 2(X1 −X2)2
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?
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n