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 so-called Pearson system.
The normal distribution is also a special case of the chi-squared distribution, since making the substitution
![]() |
(64)
|
gives
![]() |
![]() |
![]() |
(65)
|
![]() |
![]() |
![]() |
(66)
|
Now, the real line is mapped onto the half-infinite interval
by this transformation, so an extra factor of 2 must be added to
, transforming
into
![]() |
![]() |
![]() |
(67)
|
![]() |
![]() |
![]() |