Suppose that X is a discrete random variablewith the follo wing probability mass fundion:23p(x) | 20/3 | 0/32 (1 - €) (3 T(1-8)/3where OLeLI is a parameter. the following Rindependent observations were taken from suchdistribution: (3,0, 2,1,0,3,0,2,I, 0, 2, 1). Use themethod of maximum likelihood to find the estimateof o.

Given X 0 1 2 3 Frequency 4 3 3 2

Answered: Suppose that X is a discete 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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a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
Suppose that discrete random variable X~uni[1,2,3] A random sample of n=36 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.5, assuming that the sample mean would be measured to the nearest tenth.
The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places. Then, Apply the central limit theorem for the binomial distribution to find the probability that the number of individuals in Lance's sample who regularly skip breakfast is less than 9898. Express the result as a decimal precise to three places.
Let Z be a random variable with E(Z) = 12 and Var(Z) = 5. Based on the statement above, determine which of the following statements are true and false. Show complete solution. a) E(3Z + 10) = 46 b) E(Z^2) = 160 c) Var(10)=10
Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
You have two random variables M and N where the joint pdf is given by f_M,N(m,n) = 1/2 if m + n >=2, m <=2 and n <=2 and 0 o/w. What is the cdf?
A study of the wages of 100 persons grouped in class-intervals of Dollar 10-12, 12 14 ... etc. revealed the mean wage to dollar. 32.02 and its standard deviation as dollar. 13.18. Later, it was discovered that the wage of a labourer who was getting Dollar 57 was misread as 27. Calculate the correct mean and standard deviation?
X is a normally normally distributed variable with mean u =10 and standard deviation a =4. Find A) P(x 1) C) P(10
Let Y1, Y2, Y3 denote independent exponential random variables with respective rates A; for i e {1,2, 3}. Answer the following questions; (a) Compute E[Y1 +Y2 +Y3]Y1 > 1, Y2 > 2, Y3 > 3]. Your answer should be an expression using A;'s. (b) Compute the probability that Yı is the smallest among three random variables. Hint: P(X < Y) = S" Px,y (x, y)dædy where Px,Y (x, y) is the joint density function.
x = 1,2,...,9. (This 2. Let X be a discrete random variable with pmf f(x) = log₁0 distribution, known as the Benford distribution, has been shown to accurately model the probabilities of the leading significant digit in numbers occurring in many datasets.) a) Verify that f(x) satisfies the conditions of a pmf. b) Find the cdf of X.

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