8.7 In this problem we refer to Example 8.6. Suppose the random variables (X1, Y₁), (X2, Y2),..., (Xn, Yn)are all independent and that for each i we have X₁, Yį distributed as N(μ₂, 0²).a) Set vand express the likelihood function in terms of the parametersH1,Hn, and v.b) Show that the log likelihood is11(μ₁,, μn, v) = 2n log - +nlogv-v2πni=1(Xi − µi)² + (Yi - fi) ²2c) Calculate the MLEs for the parameters 1,,fln, V.

From the given information, X1,Y1, X2,Y2....Xn,Yn are all independent and that for each i, and…

Answered: .7 In this problem we refer to Example…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

3. Let X1, X2, ..., Xn be a random sample from the population N(µz,0²) and Y1, Y2,...,Ym a random sample from the population N(µy, o2), where both means (z and Hy) are assumed known. Note that the two distributions have a common variance of o². (a) If the X's and Y's are independent, show that the MLE for the common variance is: E (Xi - H2)² +E(Y; - Hy)? n+ m (b) Is the MLE found above unbiased for o2?
(36) Let X be a random variable with p.d.f. tet x= 1,2,3 f(x) = find (1) k (2) E(x) O.w
1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).
Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
4. Estimation based on a function of the observation. Let e be a positive random variable, with known mean p and variance o?, to be estimated on the basis of a measurement X of the form X = vOW. We assume that W is independent of e with zero mean, unit variance, and known fourth moment E[W4]. Thus, the conditional mean and variance of X given O are 0 and 0, respectively, so we are essentially trying to estimate the variance of X given an observed value. (a) Find the linear LMS estimator of e based on X = x. (b) Let Y = X?. Find the linear LMS estimator of O based on Y = y.
Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
Chapters: Normal and Exponential random variables. Q: Let X be an exponential random variable with rate 1. If a and b are positive numbers, then P(X > a + b|X > b) = P(X > a). a. Explain why this is called the memoryless property b. Show that for an exponential rv X with rate 1, P(X > a) = e-ad.
3. Suppose X is a discrete random variable with pmf defined as p(x) = log10 ( for %3D x = {1,2,3,...9} Prove that p(x) is a legitimate pmf.
Suppose that X1, X2, ..., Xn is a random sample with a probability law P(x) = {0x, 0 Where 8 is an unknown parameter. si x = 1,2,3, 4; otherwise A. Get an estimator of 0 by the method of the moments. B. Prove that the estimator is unbiased.
The proportion of students who get an undergraduate degree in statistics in four years was compared for meals and females. Random sample of 200 male statistics majors and 200 female statistics majors were taken. Choose the appropriate parameter(s) for this situation. [Hint: The .parameter(s) to be used in your hypothesis] ule u 2.00 jo None of these .A O P1-P2.B O H.C O Hi- H2.D O P.E O إنهاء المحاولة . . . PREVIOUS ACTIVITY Note P Type here to search Ae Ps 近

Compare these three estimators. Which one do youprefer and why?

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS