Welcome to this page of essential maths' reading!  Here you will find some of the pure mathematics associated with the normal distribution in probability theory.

[1] The function .

The function is defined and positive for all values of x;

Consider now what can be obtained from differentiation.

From the above we have two behaviour tables.

So there is a single local maximum at and two points of inflexion at

This information enables us to sketch the graph as below.

The maximum at (0,1) gives a scale in the y-axis and the distance from the y-axis to the point of inflexion gives a scale to the x-axis. There are no other distinguishing features of the graph.


Let us now change the scale in the x-axis by stretching parallel to the x-axis by scale with the y-axis fixed. That corresponds to changing x to so the new graph is or more clearly written as . The consequence of this transformation is that the points of inflexion now correspond to .


Now let us consider (where k is a constant) as a probability density function. It has some of the essential properties, namely that y is defined and positive for all values of x. In addition we need that . We address that next.

First we know that So we know that there exists a such that . Hence,

Hence, is finite, and so therefore are both and finite.

Now suppose that , say.

We need three dimensions to deal with this. Take the horizontal plane as the usual xy-plane, but with an additional z-axis which is vertical.

Consider in geometrical terms what happens when the curve is rotated about the z-axis. The result is rather like an inverted bell shape. The equation of that surface will be , where r is the radial distance from the y-axis.

We have by Pythagoras' Theorem that

We consider the calculation of the volume under that surface and the xy-plane.

We can evaluate it in two different ways.

One way is to consider it built up of thin walled cylinders with axis on the z-axis and of thickness and consequent volume to first order.

Then its volume is


The other is to consider the volume built up of thin sections parallel to the xz-plane of thickness . Then the volume of such a section is to first order is

since we took earlier. So the volume generated by all such sections cumulatively in the limit as is

since y is just a dummy variable.
Taking the two ways together we have

So we conclude that

Now we return to the probability density function for which we required

Now we can substitute for the value of the integral so we have

Hence will serve as a probability density function. The corresponding random variable is known as the standardised normal variable. The significance of standardised will appear presently. Let that variable be denoted by Z.


Now we turn to the properties of the random variable Z. By symmetry the mean of Z is 0. To calculate the variance of Z we use

where we already know that

So we have

Now the square bracket contributes zero to the outcome since

The remaining integral we recognise as the one just calculated; we know that


The variance of the random variable is therefore 1, and so also is the standard deviation, its square root.

Thus we have the neat result that the distance from the origin along the z-axis to the point of inflexion is precisely the standard deviation of the random variable - a very simple and satisfying result.

The density function of the standardised normal variable is normally denoted by
and the corresponding cumulative probability function by
since there is no simple closed form for the integral function.


We can now consider a generalisation by defining another random variable
We seek the probability density function . We have that

Alternatively we have
(where the last step is justified by the substitution ). So, by comparing the two versions, we have

It follows that the density function for X is given by

By the results (or by integration)

(where A is a random variable and b, m are constants), the corresponding mean and variance for X are

X is the generalised normal variable with mean and variance . There is no call for tabulation of since for any probability
we can use the tables for the cumulative distribution function of Z with the values and .

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