Exercises 2.1 Central limit theorem We have N independent r.v. xi = f+1;????1g such that p(+1) = p(????1) = 1 2 .

We dene the sum X =

PN i=1 xi 1. Compute the mean-value E[X].

2. Compute the variance E[X2] ???? E[X]2.

3. Compute the probability P(M) such that X = M.

4. Take the limit N ! 1 ofF p

N 2 P(

p Nx) with x 2 R.

5. Deduce the result above using the central limit theorem [26] (Hint: use the Stirling formula).