3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n. 3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribu...
3. [6 pts] Let X1, . . . , Xn be a random sample frorn a distribution with variance σ2 < oo. Find cov(X, -X,x) for i = 1, ,n. 3. [6 pts] Let X1, . . . , Xn be a random sample frorn a distribution with variance σ2
3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n.
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo 5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
Let X = (X1, . . . , Xn) be a random sample of size n with mean μ and variance σ2. Consider Tm i=1 (a) Find the bias of μη(X) for μ. Also find the bias of S2 and ỡXX) for σ2. (b) Show that Hm(X) is consistent. (c) Suppose EIXI < oo. Show that S2 and ỡXX) are consistent. Let X = (X1, . . . , Xn) be a random sample of size n with mean μ...
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise 1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n). Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
2. Let Xi, X2, . Xn be a random sample from a distribution with the probability density function f(x; θ-829-1, 0 < x < 1,0 < θ < oo. Find the MLE θ
Let Xi, , Xn be a random sample from a n(o, σ*) distribution with pdf given by 2πσ I. Is the distribution family {f(x; σ), σ 0} complete? 2. Is PCH)〈1) the same for all σ ? 3. Find a sufficient statistic for σ. 4. Is the sufficient statistic from (c) also complete!? Let Xi, , Xn be a random sample from a n(o, σ*) distribution with pdf given by 2πσ I. Is the distribution family {f(x; σ), σ 0}...