The differential equation having a normal distribution as its solution is
(60)

This dynamic will be dictated by the distance from mean, the slope will be positive before mean and negative after mean and y will asymptotically reach 0.
since
(61)

(62)

(63)

It can be shown that the optimums of change in Y happen at sigma far from the mean. In other words the maximum fall rate for the probability happens at x=+ – sigma
This equation has been generalized to yield more complicated distributions which are named using the socalled Pearson system.
The normal distribution is also a special case of the chisquared distribution, since making the substitution
(64)

gives
(65)


(66)

Now, the real line is mapped onto the halfinfinite interval by this transformation, so an extra factor of 2 must be added to , transforming into
(67)

