Examples

Simpson’s Rule on the Gaussian distribution

The function is very important for probability and statistics, but it cannot be integrated analytically.

Apply Simpson’s Rule to approximate the integral:

with and . What error bound is guaranteed for this approximation?

Solution

(1) We need a table of values of and :

These can be plugged into the Simpson Rule formula to obtain our desired approximation:

To find the error bound we need to find the smallest number we can manage for .

Take four derivatives and simplify:

On the interval , this function is maximized at . Use that for the optimal :

Finally we plug this into the error bound formula: