Archive for June 21st, 2010

Monday Math 124

June 21, 2010

The central limit theorem tells us that for sufficiently large n, the mean of n samples of any random variable for which the first and second moments (and thus mean and variance) exist has a distribution that approaches, and can thus be approximated by, a normal distribution (Gaussian distribution).

Now, consider the case of the probability of getting exactly k heads in n (fair) coin flips. This is described by the binomial distribution with p=q=1/2
,
where
.
The mean is .

Now, suppose we have 2n flips (so that the mean is n), and we consider the probability of getting  heads (so that x is the difference from the mean). What is the Gaussian approximation for the probability distribution when n is large, and what are the limits on x for which the approximation is valid?
Solution: