Euler’s e
The shape of the
normal distribution is mainly due to the transcendental number e, symbolized as
ε
(epsilon) by
Euler (1707-1783) in his 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 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 … Rounded to the two decimal points, e equals 2.72. If you
would lend a dollar at the 100% of interest rate computed continuously, 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.
Increasing the
frequency of compounding to infinitely small time intervals, defines the e as
the limit of the function
which is,
approximately, 2.72. Using the scalar module of the Visual Statistics Studio
or, directly, by
unfolding the scalar display
The e can be also
defined as the sum of the infinite series
or the series of gamma functions
When the argument
of a Gamma function is an integer, the gamma function is just the factorial
function offset by one,
Continuously
compounded interest is a prototype of the continuous growth. Suppose a
bacterial culture rapidly increases and that adding penicillin reverses the
process. The
growth and decline of a bacterial culture (of 1,000 bacteria) was plotted as
shown in the figure below. After an initial growth the trend was reversed by
addition of penicillin.
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Accelerated and Decelerated
Growth Function
In
the Visual Statistics Studio select
( Functions, Sketch-a-Graph ) define
abscissa as 0 to 1 (growth) and plot exp(X^2) where exp stands for the e to the
power in the parentheses. Invoke another
graph and plot the same function for the -1 to 0 interval (decline) and
juxtapose both graphs, as
Next,
define the abscissa as -1 to 1 and plot the y = exp(x^2) function, as (decline,
growth)
Try
whether the reciprocal of the exp(x^2) function will change the function to the
(growth, decline) form
This
reciprocal can be written in algebraic notation as
If
you do not know why, select ( Operations, Exponentiate Variables )
Now,
extend the abscissa into the -3 to 3 interval, characteristic of the standard
scores, as
The
y = exp( -.5 * x^2) function resembles the standard normal distribution, which
is defined as a special case of the Euler’s gamma density functions
as
and
can be plotted as
Why
the a coefficient of the Euler’s
gamma density function equals .3989 will be discussed in one of the following
chapters.