Let X be a Possion (X) random variable. We have seen in class that E[X] =...

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Let X be a Possion (X) random variable. We have seen in class that E[X] = Var(X) = X. Suppose that we do not know the true value of X and want to estimate it from observed data {1,2,...,n}. There are two possible ways to do estimate X: (1) use the sample mean * = Η Σ 1 2, and (2) use the sample variance S² = (x₁ - x)². Please note that in sample variance, the denominator is n - 1 instead of n. In this assignment, you will compare the two estimators. In the following questions, we assume that λ = 10. 1. Generate n = 10 independent Poisson (X) random variables, calculate the sample mean. Do the above 1000 times, then you have 1000 observations of the sample mean (each of them is calculated from n = 10 independent Poisson (A) random variables.) Generate the boxplot and histogram of the 1000 observation of sample means. 2. For n = 10, repeat Part 1 with the sample variance. 3. Compare the boxplot and histogram you obtained from Part 1 and 2. Comment on the difference between them. (Hint: range? skewness? IQR? etc.) 4. Repeat Part 3 with n = 100 and n = 1000. What trend do you observe? 5. For n = 100, using the central limit theorem, find an appropriate normal pdf to approximate the histogram for 1000 sample means. Overlay the pdf you choose onto the histogram. Let X be a Possion (X) random variable. We have seen in class that E[X] = Var(X) = X. Suppose that we do not know the true value of X and want to estimate it from observed data {1,2,...,n}. There are two possible ways to do estimate X: (1) use the sample mean * = Η Σ 1 2, and (2) use the sample variance S² = (x₁ - x)². Please note that in sample variance, the denominator is n - 1 instead of n. In this assignment, you will compare the two estimators. In the following questions, we assume that λ = 10. 1. Generate n = 10 independent Poisson (X) random variables, calculate the sample mean. Do the above 1000 times, then you have 1000 observations of the sample mean (each of them is calculated from n = 10 independent Poisson (A) random variables.) Generate the boxplot and histogram of the 1000 observation of sample means. 2. For n = 10, repeat Part 1 with the sample variance. 3. Compare the boxplot and histogram you obtained from Part 1 and 2. Comment on the difference between them. (Hint: range? skewness? IQR? etc.) 4. Repeat Part 3 with n = 100 and n = 1000. What trend do you observe? 5. For n = 100, using the central limit theorem, find an appropriate normal pdf to approximate the histogram for 1000 sample means. Overlay the pdf you choose onto the histogram.

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1Histogram and Box plot of the 1000 observations of the sample mean n10 2 Histogram and Box plot of ... View the full answer