Editing Pascal's triangle (section)

== Relation to binomial distribution and convolutions ==
When divided by <math> 2^n</math>, the <math> n</math>th row of Pascal's triangle becomes the [[binomial distribution]] in the symmetric case where <math> p = \tfrac{1}{2}</math>. By the [[central limit theorem]], this distribution approaches the [[normal distribution]] as <math> n</math> increases. This can also be seen by applying [[Stirling's formula]] to the factorials involved in the formula for combinations.

This is related to the operation of [[discrete convolution]] in two ways. First, polynomial multiplication corresponds exactly to discrete convolution, so that repeatedly convolving the sequence <math> \{ \ldots, 0, 0, 1, 1, 0, 0, \ldots \}</math> with itself corresponds to taking powers of <math> x + 1</math>, and hence to generating the rows of the triangle. Second, repeatedly convolving the distribution function for a [[random variable]] with itself corresponds to calculating the distribution function for a sum of ''n'' independent copies of that variable; this is exactly the situation to which the central limit theorem applies, and hence results in the normal distribution in the limit. (The operation of repeatedly taking a convolution of something with itself is called the [[convolution power]].)