1. Let X1, X2, .., X, constitute a random sample from probability density functiongiven byf(x; 8) = }0z(0 – x), 0 2. Let X,, X2, ... ,X, constitute a random sample from probability density functiongiven byxª-!e, x>0f(x;a,6) = {(r(a)0<,elsewhereWhere a > 0 is known.a. Find the MLE of 0.b. Find the expected value and variance of ô.

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Answered: 1. Let X, X2, - ,X constitute a random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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3. Consider a pdf f so that a random variable X~ f has expected value E[X] = 3 and variance V[X] = 10. Now consider n = 10 i.i.d. random variables X1, X2,..., X10 drawn from f. Let X = 10 (a) What is E[X]? (b) What is VIX]? (c) What is the standard deviation of X? (d) Which is larger PX > 4] or P[X > 4]? 10 ₁=1 Xi.
Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the above
5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?
b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
The joint pdf of two variables X and Y is fxy(x, y) = X and Y are independent random variables. 1 e-(x/6+y/3) for x ≥ 0, y ≥0. Show that 18
b) The probability density function of random variable X is given as: - Вх {*5 if 0 < x < 1 otherwise f(x) = If the expected value of the random variable X is 1/3. Find the values A and B.
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If pdf of a random variable is given by fx(x) = e* for x20 find My(v), m, and m,. X
b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.

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1. Let X, X2, - ,X constitute a random sample from probability density function given by f(x;8) = 02 (0 – x), 0