COMBINATORIAL OMEGA DISTRIBUTION FUNCTIONS:
A Brief Introduction
Rutgers Business School
A Handful of Coins
Consider deliberately dropping 10 pennies and asking your 5-year old niece, Lulu, to keep those that come up "heads." This is a game you & Lulu play regularly. Being a precocious child, she knows the following:
f(2) = Prob(Y=2) = (½)ış10!/(2!8!) = 45/1024 = 4.4%
f(y) = Prob(Y=y) = (½)ış10!/(y!10-y!)
Lulu's winnings can be thought of as the sum Y = x1+... + xn . The variables (x1 ...xn) are zeroes or ones depending on whether the respective coins come up "tails" or "heads." The frequency/probability distribution of Y looks like this:
Symmetrical binomial frequency function for n=10
The numbers on the bars are simply the values of 10!/(y!10-y!). We call this a frequency distribution. To obtain a probability function, Lulu divides these values by 210 =1024.
Aha, Lulu says, but what if the coins are not pennies but of arbitrary denomination?
Suppose that Lulu's 10 coins include nickels, dimes, and quarters in the following combination:
5, 5, 5, 10, 10, 10, 25, 25, 25, 25
Lulu's take is now the weighted sum Y = 5x1+5x2+... + 25x10. Its distribution f(y) = Prob(Y=y) is no longer binomial. It is an
omega function, a generalization of the binomial for weighted sums. For this set of coins it has a mean of 72.5 and looks like this:
It is very likely that Lulu will walk away with 70 or 75 cents from a single play of this game. The chance of larger or smaller amounts diminishes. For example, the chance of getting all the coins ($1.45) is less than 1/10 of 1%.
If you were to add 3 pennies to your coins, you'd have 1, 1, 1, 5, 5, 5, 10, 10, 10, 25, 25, 25, 25 as the parameters of f(y).
Here is how the new distribution looks:
After a while, Lulu will no longer be nickel-and-dimed!
She is an unconventional kid that demands unconventional coins. So, you reach into your virtual deep pocket and dazzle her with a fantastic collection of coins denominated 18, 7, 3, 9, 5, 6, 21, 30, 27, 15, 61, 12, and 24 cents, respectively. You toss them and again Lulu keeps all the "heads."
If the coins are fair, in the conventional sense, do you have any idea what the shape of the distribution would be?
First, try to imagine and then click here.
This is just a small indication of what an omega function might look like. The actual variety of omega forms is, in fact, is unimaginable. The second Reference, below, contains some additional specimens.
More will eventually be provided. In the meantime, contact me, if interested.
R e f e r e n c e s / S o u r c e s
Papayanopoulos, L., Generalized Variance of Multivariate Omega Functions and Duality,
Annals of Operations Research, 116 (2002) 21-40.
Papayanopoulos, L., Application of the Weighted Binomial in Quality Control,
Nonlinear Analysis, Vol. 30:7 (1997) 4025-4032.
Papayanopoulos, L., Preventing Minority Disenfranchisement Through Dynamic Apportionment
of Legislative Power, Ann. of Dynamic Games, Vol 1 (1994) 386-394.
Papayanopoulos, L., Properties and MIS Applications of Combinatorial Distribution Functions,
Journal of Management Information Systems, Vol. 2, No. 1 (1985), 77-95.
Papayanopoulos, L., Duality in an LP with Bounded Variables,
Advances in Management Studies, Vol. 2, No. 2 (1983), 153-171.
Papayanopoulos, L., COMBINA, Columbia Courseware Series, IBM/RD Irwin, Inc.
(limited distribution PC Software) (1986).