Law of Large Number - Central Limit Theorem - Econometrics Assignment Help

Assignment Task:

Part 1: Law of Large Number

The first part of the data analysis serve the purpose to get you familiarize the LLN. LLN characterize the following fact:

Suppose {Xn} is a sequence of random variables draw from an underlying population space, then the sample mean of {Xn} defined as Xn = 1 n Pn i=1 Xi, converge in Probability or Almost Surely to the population mean of that underlying population space.

(a) • Randomly draw 10000000 samples from from N(4,16), plot the histogram of the samples and underlying population distribution together

• What is the theoretical expectation for the population space? Variance? and Standard Deviation?

• Let sample size n equal to 1, 2, 4, 8, 16, · · · , 2 22 = 4194304, 2 23 = 8388608 respectively. Randomly choose n samples from the original 10000000 sample space. (This is for speed consideration, you could draw n samples from N(4,16) directly each time, but that would cost a lot of time), compute the sample mean Xn for each n.

• Plot the sample mean with each sample size together with the population mean. What is your conclusion from the graph.

• Change the parameters of the underlying distribution N(4, 16) to whatever you like, follow the same procedure again. Can you draw the same conclusion?

(b) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by Binomial Distribution B(50, 0.4)

(c) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by t-Distribution t(10). When you change your own parameters as the last bullet sub question asked, make sure the degree of freedom be greater than 2, otherwise the sample mean may not converge, think about why?

(d) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by F-Distribution F(9,7). When you change your own parameters as the last bullet sub question asked, make sure the degree of freedom df2 be greater than 4, otherwise the sample mean may not converge, think about why?

Part 2: Central Limit Theorem

The second part of the data analysis serve the purpose to get you familiarize the CLT. CLT characterize the following fact:

Suppose {Xn} is a sequence of random variables draw from an underlying population space, then the sample mean of {Xn} defined as Xn = 1 n Pn i=1 Xi, converge in distribution to Normal Distribution N(µ, σ2 ), where µ = E[Xi ], σ2 = Var[Xi] n

(a) • Randomly draw 10000000 samples from from N(20,25), plot the histogram of the samples and underlying population distribution together

• What is the theoretical expectation for the population space? Variance? and Standard Deviation?

• Let sample size n equal to 10 and 100000 respectively. We want to conduct the following experiment on both large sample cases (n = 100000) and samll sample cases (n = 10). Repeat the following process for 2000 times: Randomly choose n samples from the original 10000000 sample space. (This is for speed consideration, you could draw n samples from N(20,25) directly each time, but that would cost a lot of time), compute the sample mean Xn for each n. As a result, we will have 2000 X10 and 2000 X100000, write down the first and last 10 of them respectively.

• Plot the histogram of 2000 sample mean X10 together with N(20, 25 10 ). What is your conclusion from the graph.

• Plot the histogram of 2000 sample mean X100000 together with N(20, 25 100000 ). What is your conclusion from the graph.

• Plot the histogram of 2000 normalized sample mean X q10−20 25 10 together with N(0, 1). What is your conclusion from the graph.

• Plot the histogram of 2000 normalized sample mean X q100000−20 25 100000 together with N(0, 1). What is your conclusion from the graph.

• Change the parameters of the underlying distribution N(20, 25) to whatever you like, follow the same procedure again. Can you draw the same conclusion?

(b) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by B(40, 0.2).

(c) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by t-Distribution t(10). When you change your own parameters as the last question asked, make sure the degree of freedom be greater than 2, otherwise the sample mean may not converge in Normal Distribution, think about why?

(d) Answer the same bullet sub-questions of (a) while replacing the underlying distribution by F-Distribution F(8,6). When you change your own parameters as the last question asked, make sure the degree of freedom df2 be greater than 4, otherwise the sample mean may not converge in Normal Distribution, think about why?

Part 3: Extra Credit

Find a new distribution, draw samples from that distribution and follow the same procedure in the previous two parts. See if LLN and CLT still hold.

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