1. (20) Suppose X1,..., X, are i.i.d. with commonprobability density function:fo(x) =*€ (-1,1),Ger2sinh(0)'where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter.In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1if Xi > 0 and Yi0 otherwise.(a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn-(Give the equation only, no need to solve ÔMLE. X).(b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn.(Give the equation only, no need to solve мOM, X).(c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi-mize the likelihood function of Y₁...., Yn).(d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.

Part (a) (a) Give an equation for the maximum likelihood estimator θ^MLE,X based on X1,...,Xn. We…

Answered: 1. (20) Suppose X1,..., X, are i.i.d.…

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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Suppose that the unknown X is ≥ 2 and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?
Suppose that X and X, have joint probability density function x2, if 1< xı < 2 and 1 < r2 < 3 10, fx1.x.("1, 2) = otherwise. What is the joint probability density function fy,y, of Y1 = X1/X2 and Y, = X2?
Find the average lifetime of the machine part. Calculate the probability that the lifetime of the machine part is less than 5.
3. The probability density functions of two statistically independent random variables X and Y are fx(x) = }u(x - 1)e-lx-1)/2 fr(y) =u(y- 3)e-(s-3)/4 %3D Find the probability density of the sum W =X+Y.
2. Let Y₁, Y2,..., Yn denote a random sample from the probability density function (0+1)yº; 0 −1 otherwise. f(y;0) = { 0, (a) Find the MLE for 0, say . Be sure to check the second derivative. (b) Find the method of moments estimator of 0, say . (c) Suppose that 1000 'yi = 750.7516 and 1000In(y) = -329.0056, calculate 8 and 0. i=1
Please try to solve complete in one hour
The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…
suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)
Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is: (image) Find the value of k that makes f(x) a legitimate probability density function, and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.
Let X will be continuous random variable with probability density function as below. f(x) = {4x , (0
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

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