b)Let X₁, X2, X3,..., X30 be a random sample of size 30 from the population X with the momentgenerating function given byMx (t) = ez(et-1)Suppose that Y = Σ201 X₁,i)Determine the probability density function of Y and find P(6 < Y < 10).ii) Use the central limit theorem to compute P(6

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Answered: b) Let X₁, X2, X3,..., X30 be 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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Let X₁, X2, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: I e) f) g) where k> 0 is a constant. f(x; 2) = 2-kxk-1e (k-1)! if x > 0 elsewhere " Justify whether or not Â is a uniform minimum variance unbiased estimator of 1. Check whether or not the MLE of λ is a consistent. Suggest with a Fisher's factorization theorem, the sufficient estimator of 1.
b) Let X₁, X2, X3.....Xn be a random sample of n from population X distributed with the following probability density function: ze zo, f(x;0)=√2m0 0, (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. if -∞
Let X₁, X₂,..., X, be independent, uniformly distributed random variables on the interval [0, b]. a) Find the probability distribution function of X(n) = max(X₁, X₂,..., Xn). Fx (t) = for 0 ≤t≤ b and zero elsewhere I b) Find the probability density function of X(n)- fx (t) = c) Find the expected value of X(n) E(X(n)) = for 0 ≤t≤ b and zero elsewhere
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.
Student 1:33 PM Wed Apr 21 * 73% O O Q B STA+2023+EC+C12 Home Insert Draw Layout Review View Calibri Bold (Body В I B I U А.. A Read Only - You can't save changes to this file. Save a copу 3. Are health and happiness related? The following data represents the level of happiness and level of health for a random sample of individuals from the General Social Survey. Does the evidence suggest that health and happiness are related. Let a=0.05 Education/Health Excellent Less than High Good Fair Рor 72 202 199 62 School High School Junior College 465 877 358 108 80 138 49 11 Bachelor 229 276 64 12 Graduate 130 147 32 2 ||
b) Let Y1, Y2.,Y5 be independent random variables with probability density function y 1 e 4 4 f(y)= ,y > 0 ,elsewhere 3 Determine the distribution and parameter of V = EY; using the method of moment i=1 generating function. Hence, find the mean of V using the moment generating function.
Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fx,y(x,y) of X and Y on its support.
a) Let Y be a random variable whose probability density function and cumulative distribution function is given by b) and, f(y;0) = ey-8 (1+ey-8)² F(y; 0) = 1- " 1 1+ey-8, -∞ 1. ii) Find the power of the test which rejects Ho if Y > 1. iii) If a test that rejects Ho given Y₁ ≥c is considered. What will be the value of c when the level of significance of the test is 0.1. The daily salaries (in rands) for an international bank are normally distributed with mean and standard deviation of 14.20 and 3.50, respectively. Suppose that the bank randomly selected fourty (40) tellers and hire them, paying each teller on average of 12.20 per day. Test at 1% level of significance whether the bank can be accused of paying poor salaries or not?
Let Y₁, Y₂,..., Yn be a random sample from the inverse Gaussian distribution with probability density function: f(y, μ, 2) = { Where μ and are unknown. a) What are i. 1 -ACT √ λ 2ny3)že elsewhere if y > 0 the likelihood ii. the log likelihood functions of u and λ. b) Find the maximum likelihood estimators of u and 2.
Let Z₁ = X₁-μx ~N(0,1), and Wi σχ = Yi-HY~N(0,1), for i = 1,2,3,...,10, then: oy i) State, with parameter(s), the probability distribution of the statistic, T = ii) Find the mean and variance of the statistic T = Σt, wi? Σ, Z² Σt=1Zi 4i=1 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of such that P(T > B) = 0.01, where T = Σ₁₁Z₁² + 10₁ W₁². ẞ Σ{1Z2 Σi,

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b) Let X₁, X2, X3,..., X30 be a random sample of size 30 from the population X with the moment generating function given by Suppose that Y = Σ1 Xi, Mx (t) = e(e¹-1) i) Determine the probability density function of Y and find P(6 < Y < 10). ii) Use the central limit theorem to compute P(6