The most important function in probability and statistics is the density function for the standard normal distribution, which is the familiar bell-shaped curve. The function is

Use Simpson’s rule with n = 20 for the following.
(a) The area under this curve between x = -1 and x = 1 represents the probability that a normal random variable is within 1 standard deviation of the mean. Find this probability.
(b) Find the area under this curve between x = -2 and x = 2, which represents the probability that a normal random variable is within 2 standard deviations of the mean.
(c) Find the probability that a normal random variable is within 3 standard deviations of the mean.

f(x) 1 2TT =e-x12