1.
Matlab
all the terms should be dened, and anyone reading it
Law of Large Numbers and the Central Limit Theorem:
Generate N random variables that are uniformly distributed on [0,1] and nd the average (e.g., in
Matlab you can use sum(rand(1,N))/N). Is this number close to the expected value of a uniform
random variable? Plot this average versus N, and observe whether it converges to the expected
value. Now do the same as above, except we subtract the mean, and normalize with N instead of
N, so that you have sum(rand(1,N)-0.5)/sqrt(N). Do this many (say M) times, which will give
you M numbers at the end. What distribution should these number follow if N is large? Draw a
histogram of these M numbers using the hist command to get a clue. Comment on the reason.
What is the eect of N on your conclusion. Say, if N = 2 would this work? What about N = 10?
Solution
Matlab
all the terms should be dened, and anyone reading it
Law of Large Numbers and the Central Limit Theorem:
Generate N random variables that are uniformly distributed on [0,1] and nd the average (e.g., in
Matlab you can use sum(rand(1,N))/N). Is this number close to the expected value of a uniform
random variable? Plot this average versus N, and observe whether it converges to the expected
value. Now do the same as above, except we subtract the mean, and normalize with N instead of
N, so that you have sum(rand(1,N)-0.5)/sqrt(N). Do this many (say M) times, which will give
you M numbers at the end. What distribution should these number follow if N is large? Draw a
histogram of these M numbers using the hist command to get a clue. Comment on the reason.
What is the eect of N on your conclusion. Say, if N = 2 would this work? What about N = 10?