Thank you! 5. Let S2 be calculated from a random sample X1,..., Xn with Var(Xi) =...
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) = {pos, li - j| > 1. Let X and S2 be the sample mean and variance, respectively. a. Show that X is a consistent estimator for u. b. Is S2 unbiased for 02? Justify. - c. Show that S2 is asymptotically unbiased for 02.
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
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)...
Please give detailed steps. Thank you. 5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
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,...
2. (18 marks) Suppose that X1, ..., Xn constitute a random sample of size n from a population XN (4,0). Define n n 1 S2 (Xx - X)2 and S. (X-X) n-1 k=1 (2a) Compare S2 and Sas two estimators of o2; (25) Show that S2 satisfies the inequality: var(S2) > 1/nI(o), where I (a) is the Fisher's information function given by I(oP) = -E 1 (2 - )2 a(02)2 ex 202 02 (20) We have known the fact that...
Let X1, . . . , Xn be a random sample from a normal distribution, Xi ∼ N(µ, σ^2 ). Find the UMVUE of σ ^2 .
3. Let X1, X2, ,Xn be a random sample from N(μ, σ2), and k be a positive integer. Find E(S2). In particular, find E(S2) and var(s2).
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points