Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a...
1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with p.d.f. Fe 202, for – V2110 <x< . Assume that o = 2 is known. a) (10 points) Derive the 90% confidence interval for u that has the shortest length. You must show all details including the pivot you use. b) (8 points) Show that the sample mean is an efficient estimator for u. Assume in (c)- (f) that the prior distribution of u...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
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).
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2 1, u is unknown. Find the MLE of a. b. [10 pts If o2 = 1, p is unknown. f = X is an estimator of u. What is the MSE of this estimator? Now assume o2 is unknown. The following data is a set of observations of X1,..., Xn. Use the dataset to answer (c), (d) and (e) 11 8 9 7 6...
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...
1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independent and identically distributed sample from 2. 1 f(ru,02) x < 00, -00 < u < 00, o20 - 00 27TO2 (a) Derive the sample variance, S2, for this random sample (b) Derive the maximum likelihood estimator (MLE) of u and o2, denoted fi and o2, respectively (c) Find the MLE of 2 (d) Derive the method of moment...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
(2) Let X, X, be a random sample from normal distribution N (,o2), stribution N(u, a and let S2 be the sample variance: (a) [8pts] show that ES-g? (b) [8pts] For a random sample of size 2 (i.e. n 2), derive that /02 ~ Z2 where Z has standard normal distribution.
Let X1....,Xn be a sample of size n from a distribution with expectation u and variance sigma^2 and let u = (2X1+X2+...+Xn-1+2Xn)/(n+1) be an estimator for u. u is consistent,asymptotically unbiased ,unbiased?