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According to the Central Limit Theorem, if X1 through XN are independent and identically distributed random variables with mean M and variance S, and the random variable X=(X1+…+XN)/N is their average, then (X – M)/(S/sqrt(N)) approaches a Normal(0, 1) distribution as n goes to infinity.
1) Using Excel or MATLAB, generate 100 instances of the variable X=(X1+X2)/2, where each Xi is distributed as Bin(5,0.2) and N=2; then, create a histogram of the 100 resulting values (X-1)/(0.16/sqrt(2)), where M=1 (or n*p for the Bin(n,p) distribution) and S=0.16 (or n*p*(1-p) for the Bin(n,p) distribution).
2) Now, generate 100 instances of the variable X=(X1+…+X20)/20, where each Xi is again distributed as Bin(5,0.2) and N=20; then, create a histogram of the 100 resulting values (X-1)/(0.16/sqrt(20)).
3) Compare your histograms from steps 1 and 2; which one appears to be closer to a Normal(0,1) distribution? What would you expect to happen if you next created a histogram for 100 instances of the average X=(X1+…+X1000)/1000?
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