Homework Help is Here – Start Your Trial Now!
Math
Statistics
Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?

Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question
Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?
Transcribed Image Text:Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?
.::: F5 Below is the famous Central Limit Theorem stated in what might be called the "standardized" form: Xn is an Theorem 4.1 (The Central Limit Theorem) Suppose X1, X2, i.i.d. sample of observations of a random variable X with mean u and standard deviation o. Then for all x ER lim P(√n- 84x (X) MacBook Air F6 & AA 6 7 Page (12) σ where denotes the cumulative distribution function of a standard normal ran- dom variable. F7 μ) To put it succinctly, if the sample size is large, for an i.i.d. sample the sample mean (or the sample sum) is approximately normal. The question that always arises is how large is "large"? There are the standard rule-of-thumb answers, e.g. having at least 30 observations, which one should take with more than a couple grains of salt. First of all, the answer depends on how good an approximation is demanded by the nature of the intended application. Secondly, it depends on how close the distribution of the random variable being observed is to a normal distribution. * < x) = Φ(x) To illustrate this latter point, consider that the random variable in question is distributed uniformly on the interval (0, 1). The probability density function is not particularly close to that of a normal random variable; but at least it's a 8 6 of 9 DII F8 ... O F9 ) 0 F10 с 1:
Transcribed Image Text:.::: F5 Below is the famous Central Limit Theorem stated in what might be called the "standardized" form: Xn is an Theorem 4.1 (The Central Limit Theorem) Suppose X1, X2, i.i.d. sample of observations of a random variable X with mean u and standard deviation o. Then for all x ER lim P(√n- 84x (X) MacBook Air F6 & AA 6 7 Page (12) σ where denotes the cumulative distribution function of a standard normal ran- dom variable. F7 μ) To put it succinctly, if the sample size is large, for an i.i.d. sample the sample mean (or the sample sum) is approximately normal. The question that always arises is how large is "large"? There are the standard rule-of-thumb answers, e.g. having at least 30 observations, which one should take with more than a couple grains of salt. First of all, the answer depends on how good an approximation is demanded by the nature of the intended application. Secondly, it depends on how close the distribution of the random variable being observed is to a normal distribution. * < x) = Φ(x) To illustrate this latter point, consider that the random variable in question is distributed uniformly on the interval (0, 1). The probability density function is not particularly close to that of a normal random variable; but at least it's a 8 6 of 9 DII F8 ... O F9 ) 0 F10 с 1:
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS