aez-notes

Moldy Gelatin

Airborne spores produce tiny mold colonies on gelatin plates in a laboratory. The many plates average 3 colonies per plate. What fraction of plates has exactly 3 colonies? If the average is a large integer m, what fraction of plates has exactly m colonies?

Since the colonies are tiny and there are many plates we can safely use a Poisson distribution for the number of colonies per plate. To consider the large m behaviour we can use Stirling's approximation.

f(x) := pdf_poisson(x,x);
print(f(x));
/*
 x   - x
x  %e
--------
   x!
*/

stirling(n) := n * log(n) - n + log(2 * %pi * n) / 2;

approxLogF(x) := (x * log(x) - x) - stirling(x);

approxLogF2(x) := 1 / sqrt(2 * %pi * x);
print(float(approxLogF2(x)));

/*
0.3989422804014326
------------------
     sqrt(x)
*/


vals(x) := [x, float(f(x)), float(approxLogF2(x))];
print(vals(2^1), vals(2^2), vals(2^3), vals(2^4));
/*
[2, 0.2706705664732254, 0.2820947917738781]
[4, 0.1953668148131645, 0.1994711402007163]
[8, 0.1395865319505969, 0.141047395886939]
[16, 0.09921753162215583, 0.0997355701003581]
*/