Based on the given information,
*Correction in the above image :
Proof this Theorem 2.1 Let X1,...,X, be a random sample from a population with mean J...
Let X1, X2, ..., X8 be a random sample of size n=8 from a normally-distributed population whose mean is 7.9 and variance is 1.1. What are the mean and variance of the sample mean X? E[X] - 7.9, Var(X) 0.138 E[X] =0.988, Var(X) = 0.138 E[X] = 7.9, Var(8) = 1.1 E[X] =0.988, Var(87) - 1.1
Let x1, x2,x3,and x4 be a random sample from population with normal distribution with mean ? and variance ?2 . Find the efficiency of T = 1/7 (X1+3X2+2X3 +X4) relative to x= x/4 , Which is relatively more efficient? Why?
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Multiple Choice Question Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for je: X1 +...+ X10 Ô, 3X1 - 2X3 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to j)?|| A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
7-27. Let X1, X2,..., X, be a random sample of size n from a population with mean u and variance o?. (a) Show that X² is a biased estimator for u?. (b) Find the amount of bias in this estimator. c) What happens to the bias as the sample size n increases?
Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for ji: X1 +...+ X10 3X1 - 2X3 + 3X10 Ô1 = @2 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some distribution with mean u and finite variance. Define the sample variance S2 = (n-1)-1 _, (Xi - X)2. We will show that S2 is an unbiased estimator of the population variance Var(X1). (i) Show that ) = 0. (ii) Show that [ŠX – 1908–) -0. ElCX –po*=E-* (Šx--) == "Varex). x:== X-X+08 – ) Lx - X +2Zx - XXX - 1) + X...
. Let X1, . . . , Xn be a random sample from a population with mean µ and variance σ2 . Verify Vn = → σ2in a.s.
6) (6 pts) Let X, X, and X; be a random sample (n = 3) from a population with mean u and standard deviation o. Consider two estimators of u: T1 = (X1 + X2 + X3)/3 and T, = 0.10 X2 +0.25 X. + 0.65 X. Recall that because 71 is the sample average, E(71) - u and Var(T) = Oʻ/3. (a) (3 pts) Find the expected value and variance of T2. (b) (3 pts) Would T, or T2...
Let X1, ..., X10 be a random sample from a population with mean y and variance o2. Consider the following estimators for je: X1 + ... + X10 ë 2 3X1 - 2X3 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. B. Both estimators are biased. C. Only the second is unbiased. D. Only the first is unbiased. E. Insufficient information.