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Euler’s Gamma Distributions

The shape of the normal distribution is mainly due to the transcendental number e, symbolized as  (epsilon) by Euler in his definitive Introductio in Analysin Infinitorum, published in 1748. This constant is often Latinized as e; its origin is rather obscure, being introduced into mathematics shortly before Napier used it as a basis of his system of natural logarithms. However, it was Euler who popularized its use and named it, as many suspect, after the initial of his own name. Euler, a prolific mathematician credited with 886 published books and articles and averaging over 700 printed pages per year, used e to show the connection between exponential and trigonometric functions.

 

e: the constant of growth and decay

 

The e to a positive power is often used to describe the growth processes and to the negative power to describe the processes of decay.

 

The value of e is 2.71828182845905 and, rounded to the two decimal points, it is often remembered that if you would lend a dollar at the 100% of interest, at the end of the first year you would have $2.72.

 

To provide for a continuous growth of an investment, interest is computed continuously and added to a principal. One dollar at 100% of interest compounded annually yields two dollars. The interest compounded semiannually, at midyear, is 50 cents. Added to the principal, the amount loaned is $1.50. At the end of the year, the interest for this additional sum loaned is $.25; the total yield is $2.25. Compounded quarterly, the interest augments the principal as $1.00 + $.25 + $.31 +$.39 + $.49 = $2.44. This compounding process can be formalized by using the expansion of a series of binomials:

 

                     

 

and so on. Increasing the frequency of compounding to infinitely small time intervals, defines the e as the limit of the function

 

                                                        

 

which is, approximately, 2.718. The e can be also defined as the sum of the infinite series

 

                                              

 

Or the series of gamma functions

 

                                    

 

 

 

Continuously compounded interest is a prototype of the continuous growth, described by the exponential function

 

                                                           

 

where c is the initial size, p is the nominal growth rate given as a proportion of the unit growth rate, and t is the number of time periods. The definition of continuous decay differs from the definition of the continuous growth only by the sign of the exponent

 

                                                         

 

An illustration of the positive and negative growth may help to clarify the above discussion. Suppose a bacterial culture of 1000 bacteria increases at a rate of 30% per day. Assume that penicillin decreases the size of the culture at the same nominal rate and is added to the culture at the end of the fourth day. The rise and fall of the bacterial culture can be calculated and is summarized in the following table:

 

Time


Rate of
Change


Size of
Population

Day One

1,000e.3(0)

1,000

Day Two

1,000e.3(1)

1,350

Day Three

1,000e.3(2)

1,822

Day Four

1,000e.3(3)

2,460

Day Five

1,000e.3(4)

3,320

Day Six

3,320e.3(-1)

2,460

Day Seven

3,320e.3(-2)

1,822

Day Eight

3,320e.3(-3)

1,350

Day Nine

3,320e.3(-4)

1,000

 

Growth and Decay of Bacterial Cultures

 

The growth and decline of the bacterial culture was plotted as shown in the figure below. After an initial rapid growth the trend is reversed by addition of penicillin. At the end of the ninth day, the culture reverted to its original size.

 

 

 

Accelerated and Decelerated

Growth Function

 

Gamma distributions

Functions can be classified as algebraic and transcendental. An algebraic function is a function that is a root of a polynomial equation. A function that is not a root of a polynomial equation is called transcendental. Most of the functions that describe natural phenomena turn out to be transcendental functions as are the trigonometric, logarithmic, exponential, and hyperbolic functions. The theory of higher transcendental functions was elaborated by Euler, (1707-1783) who also introduced the beta and gamma transcendental functions. Most sampling distributions of inferential statistics belong to the family of the gamma density functions. Some textbooks on statistics ascribe the t-distribution to Student and the F distribution to Snedecor. These statisticians only called the attention to the applicability of some of the higher transcendental function to the theory of statistical inference. However, the gamma density functions are due to Euler. These functions have a general form

 

                                                          

 

Examples of gamma density functions are

 

                                                            

 

                                                          

 

                                                          

 

approximating the normal distribution which equation is

 

                                                       

 

The y1, y2, and y3 gamma functions were plotted below, as

 

 

 

Perspective on the gamma distributions

Within the statistical computer programs, the probabilities associated with the z, t F, and Chi Square ratios may be calculated by a single subroutine. This subroutine normalizes the F distribution as

 

                                                

 

After the normalization, this subroutine uses the polynomial approximations to find areas under the normal distribution, corresponding to standard z scores, as

 

                                              

 

where c1 = .196854, c2 = .115194, c3 = .000344, and c4 = .019527. The infinity is represented by a large number, usually equal to 1,000.

 

Since  F equals z-Square with (1, infinity) degrees of freedom, the probability associated with the z-Square ratio can be obtained as

 

p = fSig (1, 1000, z-Square).

 

Since F equals t-Square with (1, df) degrees of freedom, the probability associated with the t-Square ratio can be obtained as

 

p = fSig (1, df, t-Square).

 

This probability is obtained by calling the fSig subroutine as

 

p = fSig(df1,df2,F).

 

Since F equals Chi Square with (df, infinity) degrees of freedom, the probability associated with the chi square ratio can be obtained as

 

p = fsig(df, 1000, Chi-Square / df).