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Stirlings formula is an indispensable
tool for all combinatorial and statistical problems because it allows to deal
with factorials, i.e. expressions based on the
definition 1 · 2 · 3 · 4 · 5 · .... · N
:= N! |
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It exists in several modifications;
all of which are approximations with different degrees of precision. It is
relatively easy to deduce its more simple version. We have |
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ln x! |
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ln 1 + ln 2 + ln 3 + .... + ln x |
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x
S
1 |
ln y |
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With y = positive integer running from
1 to x |
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For large y we may replace the sum by an integration in a good
approximation and obtain |
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x
S
1 |
ln y |
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x
∫ 1 |
(ln y) · dy |
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With (ln y) · dy = y
· ln y y, we obtain |
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This is the simple version of
Stirlings formula. it can be even more simplified for large x
because then x + 1 << x · ln x; and the
most simple version, perfectly sufficient for many cases, results:
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However!! We not only produced a simple
approximation for x!, but turned a discrete function having values for integers
only, into a continuous function, giving
numbers for something like 3,141! - which may or may not make sense.
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This may have dire consequences. Using the Strirling formula you may, e.g.,
move from absolute probabilities (always a
number between 0 and 1) to probability
densities (any positive number) without being aware of it. |
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Finally, an even better approximation
exists (the prove of which would take some 20 pages) and which is
already rather good for small values of x, say x >
10: |
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x! |
» |
(2π)1/2 · x(x +
½) · e x |
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© H. Föll