4. Let Xi,.. . , Xn be a random sample from a distribution with the density...
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 the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval for θ. If possible find an exact CI. Otherwise determine an approximate CI. Explain your choice Let Xi., Xn be a random sample from the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval...
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).
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,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Problem 8.2 Suppose that Xi, X,.., Xn is a random sample of size n is to be taken from a population with pdf 2 In>X (In2) x We are interested in determining the approximate distribution of the sample geometric mean given by [x. If we let Y-In X, then we can re-express the geometric mean as a) Determine the mean of Y. Hint, if u = In x, then du = 1/x dx. b) Determine the variance of Y. c)...
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...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
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...