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 yaxis and the distance from the yaxis to the point of inflexion gives a scale to the xaxis. There are no other distinguishing features of the graph. [2] Let us now change the scale in the xaxis by stretching parallel to the xaxis by scale with the yaxis 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 . [3] 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 xyplane, but with an additional zaxis which is vertical.
Consider in geometrical terms what happens when the curve is rotated about the zaxis. The result is rather like an inverted bell shape. The equation of that surface will be , where r is the radial distance from the yaxis.
We consider the calculation of the volume under that surface and the xyplane. We can evaluate it in two different ways.
.
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. [4]
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
Now the square bracket contributes zero to the outcome since
The remaining integral we recognise as the one just calculated; we know that
Hence
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 zaxis 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
[5]
We can now consider a generalisation by defining another random variable
Alternatively we have
It follows that the density function for X is given by
By the results (or by integration)
X is the generalised normal variable with mean and variance .
There is no call for tabulation of since for any probability
