# According to the Central Limit Theorem, if X1 through XN are independent and ide

According to the Central Limit Theorem, if X1 through XN are independent and ide

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?