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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: