by Marco Taboga, PhD
The normal distribution is a continuous probability distribution that plays a central role in probability theory and statistics.
It is often called Gaussian distribution, in honor of Carl Friedrich Gauss (17771855), an eminent German mathematician who gave important contributions towards a better understanding of the normal distribution.
Table of contents
Why is it so important?
Main characteristics
The standard normal distribution
Definition
Expected value
Variance
Moment generating function
Characteristic function
Distribution function
The normal distribution in general
Definition
Relation between standard and nonstandard normal distribution
Expected value
Variance
Moment generating function
Characteristic function
Distribution function
Density plots
Plot 1  Changing the mean
Plot 2  Changing the standard deviation
More details
Solved exercises
Exercise 1
Exercise 2
Exercise 3
Why is it so important?
The normal distribution is extremely important because:

many realworld phenomena involve random quantities that are approximately normal (e.g., errors in scientific measurement);

it plays a crucial role in the Central Limit Theorem, one of the fundamental results in statistics;

its great analytical tractability makes it very popular in statistical modelling.
Main characteristics
Sometimes it is also referred to as "bellshaped distribution" because the graph of its probability density function resembles the shape of a bell.
As you can see from the above plot, the density of a normal distribution has two main characteristics:

it is symmetric around the mean (indicated by the vertical line); as a consequence, deviations from the mean having the same magnitude, but different signs, have the same probability;

it is concentrated around the mean; it becomes smaller by moving from the center to the left or to the right of the distribution (the so called "tails" of the distribution); this means that the further a value is from the center of the distribution, the less probable it is to observe that value.
The remainder of this lecture gives a formal presentation of the main characteristics of the normal distribution.
First, we deal with the special case in which the distribution has zero mean and unit variance. Then, we present the general case, in which mean and variance can take any value.
The standard normal distribution
The adjective "standard" indicates the special case in which the mean is equal to zero and the variance is equal to one.
Definition
Standard normal random variables are characterized as follows.
Definition Let be a continuous random variable. Let its support be the whole set of real numbers:We say that has a standard normal distribution if and only if its probability density function is
The following is a proof that is indeed a legitimate probability density function:
Proof
The function is a legitimate probability density function if it is nonnegative and if its integral over the support equals 1. The former property is obvious, while the latter can be proved as follows:
Expected value
The expected value of a standard normal random variable is
Proof
It can be derived as follows:
Variance
The variance of a standard normal random variable is
Proof
It can be proved with the usual variance formula ():
Moment generating function
The moment generating function of a standard normal random variable is defined for any :
Proof
It is derived by using the definition of moment generating function:The integral above is welldefined and finite for any . Thus, the moment generating function of exists for any .
Characteristic function
The characteristic function of a standard normal random variable is
Proof
By the definition of characteristic function, we haveNow, take the derivative with respect to of the characteristic function:By putting together the previous two results, we obtainThe only function that satisfies this ordinary differential equation (subject to the condition ) is
Distribution function
There is no simple formula for the distribution function of a standard normal random variable because the integralcannot be expressed in terms of elementary functions. Therefore, it is usually necessary to resort to special tables or computer algorithms to compute the values of . The lecture entitled Normal distribution values discusses these alternatives in detail.
The normal distribution in general
While in the previous section we restricted our attention to the special case of zero mean and unit variance, we now deal with the general case.
Definition
The normal distribution with mean and variance is characterized as follows.
Definition Let be a continuous random variable. Let its support be the whole set of real numbers:Let and . We say that has a normal distribution with mean and variance if and only if its probability density function is
We often indicate the fact that has a normal distribution with mean and variance by
To better understand how the shape of the distribution depends on its parameters, you can have a look at the density plots at the bottom of this page.
Relation between standard and nonstandard normal distribution
The following proposition provides the link between the standard and the general case.
Proposition If has a normal distribution with mean and variance , thenwhere is a random variable having a standard normal distribution.
Proof
This can be easily proved using the formula for the density of a function of a continuous variable ( is a strictly increasing function of , since is strictly positive):
Thus, a normal distribution is standard when and .
Expected value
The expected value of a normal random variable is
Proof
The proof is a straightforward application of the fact that can we written as a linear function of a standard normal variable:
Variance
The variance of a normal random variable is
Proof
It can be derived as follows:
Moment generating function
The moment generating function of a normal random variable is defined for any :
Proof
The mgf is derived as follows:It is defined for any because the moment generating function of is defined for any .
Characteristic function
The characteristic function of a normal random variable is
Proof
The derivation is similar to the derivation of the moment generating function:
Distribution function
The distribution function of a normal random variable can be written aswhere is the distribution function of a standard normal random variable (see above). The lecture entitled Normal distribution values provides a proof of this formula and discusses it in detail.
Density plots
This section shows the plots of the densities of some normal random variables. These plots help us to understand how the shape of the distribution changes by changing its parameters.
Plot 1  Changing the mean
The following plot contains the graphs of two normal probability density functions:

the first graph (red line) is the probability density function of a normal random variable with mean and standard deviation ;

the second graph (blue line) is the probability density function of a normal random variable with mean and standard deviation .
By changing the mean from to , the shape of the graph does not change, but the graph is translated to the right (its location changes).
Plot 2  Changing the standard deviation
The following plot shows two graphs:

the first graph (red line) is the probability density function of a normal random variable with mean and standard deviation ;

the second graph (blue line) is the probability density function of a normal random variable with mean and standard deviation .
By increasing the standard deviation from to , the location of the graph does not change (it remains centered at ), but the shape of the graph changes (there is less density in the center and more density in the tails).
More details
The following lectures contain more material about the normal distribution.
Normal distribution values
How to tackle the numerical computation of the distribution function
Multivariate normal distribution
A multivariate generalization of the normal distribution, frequently encountered in statistics
Quadratic forms involving normal variables
Discusses the distribution of quadratic forms involving normal random variables
Linear combinations of normal variables
Discusses the important fact that normality is preserved by linear combinations
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Let be a normal random variable with mean and variance . Compute the following probability:
Solution
First of all, we need to express the above probability in terms of the distribution function of :
Then, we need to express the distribution function of in terms of the distribution function of a standard normal random variable :
Therefore, the above probability can be expressed aswhere we have used the fact that , which has been presented in the lecture entitled Normal distribution values.
Exercise 2
Let be a random variable having a normal distribution with mean and variance . Compute the following probability:
Solution
We need to use the same technique used in the previous exercise (express the probability in terms of the distribution function of a standard normal random variable):where we have found the value in a normal distribution table.
Exercise 3
Suppose the random variable has a normal distribution with mean and variance . Define the random variable as follows:Compute the expected value of .
Solution
Remember that the moment generating function of isTherefore, using the linearity of the expected value, we obtain
How to cite
Please cite as:
Taboga, Marco (2021). "Normal distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probabilitydistributions/normaldistribution.