Let X be an exponential random variable with parameter 14. Then E[X]=4, Var(X)=16, q1=4 In(4/3)=1.49, q2=median=4ln(2)=2.614,...

 

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Let X be an exponential random variable with parameter 14. Then E[X]=4, Var(X)=16, q1=4 In(4/3)=1.49, q2=median=4ln(2)=2.614, q3=4ln(4)=6.833, and the skewness of X is 2. 1) Generate 1000 exponential random variables with parameter ¼ and store it in Exponential [100,10], so that the matrix has 10 columns and 100 rows. a. For each row, take the average of 10 entries and enter this value to the 11th column. b. Display the following descriptive statistics for all columns: sample mean, sample standard deviation, minimum, maximum, 1st and 3rd quartiles, median, and skewness. How do these statistics compare with each other as well as E[X] and the standard deviation of X? C. Generate the following charts for the 1st and 11th columns: histogram, box plot, probability plot. Compare the shape of these plots with those of exponential and normal random variables. Do you think the averages of 10 exponential random variables look like a normal random variable? Why or why not? Explain. 2) In part (1), you generated 100 samples, each having a sample size of 10. After the lab session, generate 100 samples with varying sample sizes such as 2, 5, 20, 30 and 40. Compute the sample means for each sample, and compare the shapes of the plots for all sample sizes. Can you see how the plots start depicting more and more symmetric distribution as the sample size increases, showing a clear convergence to a normal distribution? Let X be an exponential random variable with parameter 14. Then E[X]=4, Var(X)=16, q1=4 In(4/3)=1.49, q2=median=4ln(2)=2.614, q3=4ln(4)=6.833, and the skewness of X is 2. 1) Generate 1000 exponential random variables with parameter ¼ and store it in Exponential [100,10], so that the matrix has 10 columns and 100 rows. a. For each row, take the average of 10 entries and enter this value to the 11th column. b. Display the following descriptive statistics for all columns: sample mean, sample standard deviation, minimum, maximum, 1st and 3rd quartiles, median, and skewness. How do these statistics compare with each other as well as E[X] and the standard deviation of X? C. Generate the following charts for the 1st and 11th columns: histogram, box plot, probability plot. Compare the shape of these plots with those of exponential and normal random variables. Do you think the averages of 10 exponential random variables look like a normal random variable? Why or why not? Explain. 2) In part (1), you generated 100 samples, each having a sample size of 10. After the lab session, generate 100 samples with varying sample sizes such as 2, 5, 20, 30 and 40. Compute the sample means for each sample, and compare the shapes of the plots for all sample sizes. Can you see how the plots start depicting more and more symmetric distribution as the sample size increases, showing a clear convergence to a normal distribution?

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