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 distribution with variance σ2
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
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
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
3. Let X1 , X2, . . . , Xn be a randon sample from the distribution with pdf f(r;0) = (1/2)e-z-8,-X < < oo,-oc < θ < oo. Find the maximum likelihood estimator of θ.
Problem 3 Let X1, X2, ... , Xn be a random sample of size n from a Gamma distribution fr; a,B) 22-12-1/B, 0 < < (a) Find a sufficient statistics for a. (b) Find a sufficient statistics for B.
4. Let Xi, X2, ensity function f(r; , Xn be a random sample from a distribution with the probability θ)-(1/2)e-11-01,-oo <エく00,-00 < θ < oo. Find the d MLE θ