Please answer number 2 X₁, X2, Xnis a random sample from the population that followsa Poisson distribution with the density function:-0 0*f (x10) =; x = 1,2,3,....x!If you want to test Ho: 0 = 3 and Ho 0 # 3 at a significance levelof a = 5%, determine :1. Likelihood function L(x : 0) = f(x₁, x2, ...., xñ[0) and themaximum likelihood estimator for 82. The Likelihood function under Ho and H₁, is L(x: 0o =f(x₁,x2,..., xn100) and L(x: 0₁) = f(x₁,x₂,..., xn|0₁)3. Likelihood ratio statistics, namely: -2ln(x: 00, 0₁) withL (x: 00)2(x: 00, 0₁) =L (x: 01)4. test rejection area and give your conclusion if you have statisticΣ1 x = 80

Given information: Given x1, x2, x3.................xn is a random sample from the population that…

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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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A random variable Y is said to have a Singh-Maddala distribution with parameter θ > 0 if the density function of Y isf(y|θ) = 2θy(1 + y2)θ+1for y > 0. Suppose that Y1,Y2,...,Yn is a random sample from a Singh-Maddala(θ) population where θ > 0 is a parameter.(a) Verify that f(y|θ) belongs to an exponential family in traditional form.(b) Compute the likelihood function L(θ) for this random sample.(c) Determine θˆMLE, the maximum likelihood estimator of θ. You may use any method that we discussed in class for finding the MLE.(d) Find the Fisher information I(θ) in a single observation from this density.(e) Using the standard normal approximation for the distribution of a maximum likelihood es- timator based on the Fisher information, construct an approximate 95% confidence interval for θ. Note that z0.025 = 1.96.Note that the Singh-Maddala distribution is also known as the Burr Type XII distribution and is used in economics and actuarial science to model income sizes.
b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
X is a uniform random variable over the interval (3, 5). Find the density function of X for the interval (3, 5)
Let X and Y be random variables with the joint density function f(x,y)=x+y, if x,y element of [0,1], and f(x,y)=0,elsewhere. Find the expected value of the random variable Z = 10X+14Y.
Let T have an exponential distribution with parameter λ. Let Y = √T .a) Find the density of Yb) Find the expectation of Y, correct to two decimal places, for λ = 3.c) A random number generator produces uniform [0, 1] random numbers. How could you usethese to generate random numbers which have the distribution of Y ?
Let x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.
A random sampleY1 , Y 2 , ⋯,Yn is drawn from a distribution whose probability density function is given by: f (Y ) = βe − βY , Y  0 & β > 0 a). Obtain the maximum likelihood estimator (MLE) of β. b). Given that ∑ n Y i = 25 , ∑n Yi 2 = 50 , n = 50 calculate the maximum likelihood i =1 i =1 estimate of β. ( c). Using the same data as in part (b), test the null hypothesis that β =1against the alternative hypothesis that β ≠1at 5% level of significance
Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLM
1)Let x and y be two continuous random variables whose function is the probability density joint is given by : a)Draw the relationship between the variables x and y on the Cartesian axes .b)Calculate the marginal pdfs px(X)and py(Y).c)Are the v.a.s x and y independent ?
The joint probability mass function of the two random variables (X, Y) is given by f(x,y) = { 1/5(3x-y), 1<=x<=2 , 1<=y<=3 0 , otherwise• Find the P( X ≤ Y)• Find the marginal density functions of X and Y• Are X and Y independent?• Find E(XY)
Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [4.9, 5.1] mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?

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