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Q5. Central limit theorem (Exponential distribution) Suppose X1,...,X, are independent and identically distributed as the exponential distribution with the rate parameter A = 2: X1,..., Xn iid Exp(A = 2). 1. According to the central limit theorem, what are the large sample distributions of X and Z = Vī(X=1/A)? 1/A 2. Do the followings with R: a. Create a plot of the pdf of the exponential distribution Exp(A 2). b. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the sample mean X. Repeat the same procedure for 1000 times to obtain 1000 simulated sample means. Plot a histogram of the 1000 simulated sample means and calculate the mean and the SD of the 1000 simulated sample means. Comment briefly on these outputs compared to the result from question 1. c. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the standardized sample mean Z = vn(X-1/A) Repeat the same procedure for 1000 times to obtain 1000 Z's. Plot a histogram of the 1000 Z's and calculate the mean and the SD of the 1000 Z's. Comment briefly on these outputs compared to the result from question 1. 1/A

Q5. Central limit theorem (Exponential distribution) Suppose X1,...,X, are independent and identically distributed as the exponential distribution with the rate parameter A = 2: X1,..., Xn iid Exp(A = 2). 1. According to the central limit theorem, what are the large sample distributions of X and Z = Vī(X=1/A)? 1/A 2. Do the followings with R: a. Create a plot of the pdf of the exponential distribution Exp(A 2). b. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the sample mean X. Repeat the same procedure for 1000 times to obtain 1000 simulated sample means. Plot a histogram of the 1000 simulated sample means and calculate the mean and the SD of the 1000 simulated sample means. Comment briefly on these outputs compared to the result from question 1. c. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the standardized sample mean Z = vn(X-1/A) Repeat the same procedure for 1000 times to obtain 1000 Z's. Plot a histogram of the 1000 Z's and calculate the mean and the SD of the 1000 Z's. Comment briefly on these outputs compared to the result from question 1. 1/A

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Q5. Central limit theorem (Exponential distribution) Suppose X1,..., X, are independent and identically distributed as the exponential distribution with the rate parameter A = 2: X1,..., Xn iid Exp(A = 2). 1. According to the central limit theorem, what are the large sample distributions of X and Z = V(X-1/A)? 1/A 2. Do the followings with R: a. Create a plot of the pdf of the exponential distribution Exp(A = 2). b. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the sample mean X. Repeat the same procedure for 1000 times to obtain 1000 simulated sample means. Plot a histogram of the 1000 simulated sample means and calculate the mean and the SD of the 1000 simulated sample means. Comment briefly on these outputs compared to the result from question 1. c. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the standardized sample mean Z = yn(X-1/A) Repeat the same procedure for 1000 times to obtain 1000 Z's. Plot a histogram of the 1000 Z's and calculate the mean and the SD of the 1000 Z's. Comment briefly on these outputs compared to the result from question 1. 1/A
Transcribed Image Text:Q5. Central limit theorem (Exponential distribution) Suppose X1,..., X, are independent and identically distributed as the exponential distribution with the rate parameter A = 2: X1,..., Xn iid Exp(A = 2). 1. According to the central limit theorem, what are the large sample distributions of X and Z = V(X-1/A)? 1/A 2. Do the followings with R: a. Create a plot of the pdf of the exponential distribution Exp(A = 2). b. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the sample mean X. Repeat the same procedure for 1000 times to obtain 1000 simulated sample means. Plot a histogram of the 1000 simulated sample means and calculate the mean and the SD of the 1000 simulated sample means. Comment briefly on these outputs compared to the result from question 1. c. Simulate a random sample of size 100 from Exp(A = 2). Then calculate the standardized sample mean Z = yn(X-1/A) Repeat the same procedure for 1000 times to obtain 1000 Z's. Plot a histogram of the 1000 Z's and calculate the mean and the SD of the 1000 Z's. Comment briefly on these outputs compared to the result from question 1. 1/A
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