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 2*n* 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:

## Archive for June 21st, 2010

### Monday Math 124

June 21, 2010
Advertisements