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Cruise Scientific Visual Statistics Studio Statistical Distributions |
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:
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Day One |
1,000e.3(0) |
1,000 |
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Day Two |
1,000e.3(1) |
1,350 |
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Day Three |
1,000e.3(2) |
1,822 |
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Day Four |
1,000e.3(3) |
2,460 |
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Day Five |
1,000e.3(4) |
3,320 |
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Day Six |
3,320e.3(-1) |
2,460 |
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Day Seven |
3,320e.3(-2) |
1,822 |
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Day Eight |
3,320e.3(-3) |
1,350 |
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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.
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Accelerated
and Decelerated
Growth Function
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
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).