B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, withprobability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., andp € (0, 1].(i) Using the MGF (M(t) ==(ii) Find the Maximum Likelihood Estimator (MLE) for p.-(1-p)et(b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ymis an independent random sample from a Beta(02, 1) population. We want to findthe approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02.To this aim:==(i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and02 are:0₁02derive E[X] and Var[X].nΣlog(x)'Recall, that the PDF of Beta(a, b) is fy (y)00= -mΣ log(yi)=['(a) = (a − 1)! for all positive integers a and I (1) = 1)[(a+b)F(a) (b)-1(1-y)b-1 and(ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is:01=n + mE log(xi) + log(yi)

The discrete probability distribution is the distribution that has variables that take an integer…

Answered: B2. (a) Consider X₁,..., X₁ to be a…

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

(a) Let X1 ~ x² (n, 8) and X2 ~ x² (m, 1). mX1 and its parameters. nX2 (i) Find the distribution of X = (ii) If 1 = 0, name the distribution of X given in 3(a)(i).
Establish the following: 0P (E)1 , P (A u B) = P (A) + P (B) – P (A n B) , P (A) = 1 – P (A’) ii The probability that a boy with a catapult hits target A is 2/3and that he hits target B is ¾. Given the probability of hitting both targets to be 1/2, find the probability that he (a) hits at least one of the targets (b) does not hit any. iii What is the probability of having at least one 6 in three throws with a dice?
2. where The random variable, X, has a probability density function (pdf), f(x), and zero otherwise. (a.) (b) x+1, f(x) = {1x, 1 81 -1/2 < x < 0 0
Consider a random variable Xwith EJ X1=5 and o.’ = 3. Another random variable is given as Y =-8X+10. Find (a) E[X²] (b) E[XY] (c) E[Y²] and (d) 0,.
(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.
If Y = InX has a N (0, 1) distribution then E (X²) is equal to
1. Verify whether the following functions can be considered as probability mass functions: (i) P(x = x) = x² + 1 18 (iii) P(X= x) = -, x = 0, 1, 2, 3 (ii) P(X - x) - ²-2, -, x= 1, 2, 3 2x + 1 18 -, x = 0, 1, 2, 3 [Ans.: Yes) [Ans.: No] [Ans.: No]
2B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32
Let X, and X; be independent exponential random variables having pdfs f, (x) = e, x > 0 and fr,(x2) = e"", I3 > 0. Given that Y, = X,/X and Y, = X, + X2. a. Find the joint pdf of Y, and Y2 b. Find the marginal pdf of Y1.

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

B2. (a) Consider X₁,..., X₁ to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with = 0, 1, 2, 3,..., and pe (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)e derive E[X] and Var[X]. (ii) Find the Maximum Likelihood Estimator (MLE) for p.