THE GOLDEN RATIO IS IMBEDDED IN THE DERIVATIVES OF THE STANDARD NORMAL CONTINUOUS PROBABILITY DENSITY FUNCTION.

As an example lets look at the simplest of the Bell shaped curves and review some of it's characteristics.

Or it's

The first derivative.

And the second derivative.

Or it's

The value of x are the points of inflection when the second derivative is equal to zero.

The square root of two over two.

When the first derivative equals the second derivative we have for the value of x as.

Minus one and positive one half.

The value of x when the function equals the first derivative is.

Minus one half.

And when the function equals the second derivative.

Square root of three over two.

Now lets graph the three of them.

We now use the density function for the standard normal distribution with the mean = 0 and the standard deviation sigma = 1 yielding.

Or it's

The first derivative is.

And the second derivative is

Or it's

The value of x are the points of inflection when the second derivative is equal to zero.

One and minus one

The value of x when the first derivative equals the second derivative is.

Expressed in a slightly different form we have.

Its the positive reciprocal of the GOLDEN RATIO and the negative value of the GOLDEN RATIO.

Or the solution is.

and

The value of x when the function equals the first derivative is.

A minus one.

And the value of x when the function equals the second derivative is.

The positive and negative square root of two.

So lets graph all three.

So to sum it all up in a table format.

,

x =

+1, -1

-b, -a

y=

-.203695, 0.174341

0.241971

0.146763, 0.146763

After a brief introduction to the Bell Shaped curve we have shown that the Golden Ratio is imbedded in the derivatives of the standard normal continuous probability density function as well as five other famous values of x. We obtain the golden Ratio by allowing the first derivative to be equal to the second derivative of the standard normal continuous probability density function and then solving for x.

The famous variables that appear imbedded in this bell shaped curve are e, pie, plus and minus 1, The Golden Ratio and plus and minus the square root of two.

This paper was developed using mathcad 13 software by Mathsoft and then converted to HTML.

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