Normal distributions: sometimes it is useful to breast the current
The usual way of defining normal variables is to introduce the whole family of normal distributions and then to say that the standard normal is a special member of this family. Here I show that, for didactic purposes, it is better to do the opposite.
Standard normal distribution
The standard normal distribution is defined by its probability density
Usually students don't remember this equation, and they don't need to. The point is to emphasize that this is a specific density, not a generic "bell shape".
From the plot of the density (Figure 1) they can guess that the mean of this variable is zero.
Alternatively, they can look at the definition of the mean of a continuous random variable . Here the function has the shape given in Figure 2, where the positive area to the right of the origin exactly cancels out with the negative area to the left of the origin. Since an integral means the area under the function curve, it follows that
To find variance, we use the shortcut:
The total area under the curve is twice the area to the right of the origin, see Figure 3. Here the last integral has been found using Mathematica. It follows that
General normal distribution
Fix some positive and real . A (general) normal variable is defined as a linear transformation of :
Properties follow like from the horn of plenty:
A) Using (1) and (3) we easily find the mean of :
B) From (2) and (3) we have
(the constant does not affect variance and variance is homogeneous of degree 2).
C) Solving (3) for gives us the z-score:
D) Moreover, we can prove that a linear transformation of a normal variable is normal. Indeed, let be defined by (3) and let be its linear transformation: Then
is a linear transformation of the standard normal and is therefore normal.
Remarks. 1) In all of the above, no derivation is longer than one line. 2) Reliance on geometry improves understanding. 3) Only basic properties of means and variances are used. 4) With the traditional way of defining the normal distribution using the equation
there are two problems. Nobody understands this formula and it is difficult to extract properties of the normal variable from it.
Compare the above exposition with that of Agresti and Franklin: a) The normal distribution is symmetric, bell-shaped, and characterized by its mean μ and standard deviation σ (p.277) and b) The Standard Normal Distribution has Mean = 0 and Standard Deviation = 1 (p.285). It is the same old routine: remember this, remember that.