Problem 3: the central limit theorem. Consider a system comprised of N independent states o = (01, ...,ON), where each oi (i = 1, ..., N) can only take value 0 or 1. The probability of of taking value 0 is po, and the probability of of taking value 1 is p1 = 1 -po. Define the macroscopic quantity S as N S = > oi. (2) i=1 a. What is the expected average (S) and how does it scale with N?b. How does the standard deviation as for the distribution of 3 scale with N (nd only the exponent I! in as N N\")? Does it grow faster? equal? or slower as a function of N compared to (S)? c. (Extra. credit) Run the following numerical experiment: Consider N = 10? p1 = 1,14? and sample over 100 congurations in order to compute 3 for each conguration 0' : (61,... ,oN). Now increase N from N = 10 to N = 102, and then to N = 103. Plot the standard deviation of the distribution of the variable 3 as a function of N. Verify that you get the exponent in point b. Is the distribution becoming sharper (is, 05,! (S) > G as N}oo]