Properties of the Normal Distribution

Suppose that the random variable X follows a normal distribution with parameters   and  . Then the following properties hold:
1.    The mean of the random variable is

                    

2.    The variance of the random variable is :
                   Var(X) = 

3.    The shape of the probability density function is a symmetrical bell-shaped curve centered on the mean, , as shown in Figure 1.

4.    If we know the mean and variance, we can define the normal distribution by using the notation
                  

 

For our applied statistical analyses the normal distribution has a number of important characteristics. It is symmetric. Different central tendencies are indicated by differences in . In contrast, differences in  result in density functions of different widths. By selecting values for  and  we can define a large family of normal probability density functions. Differences in the mean result in shifts of entire distributions. In contrast, differences in the variance result in distributions with different widths.

Figure: Effects of   and   on the probability density function of a Normal random variable.

a.    Two normal distributions with different means.
b.    Two normal distributions with different variances and mean = 5.