Cumulative Distribution Function of the Normal Distribution

Suppose that X is a normal random variable with mean  and variance ; that is,  Then the cumulative distribution function is

                    
This is the area under the normal probability density function to the left of x0. As for any proper density function, the total area under the curve is 1; that is,

                                 
Figure: The shaded area is the probability that X does not exceed x0 for a normal random variable.