COMBINATORIAL ** OMEGA** DISTRIBUTION FUNCTIONS:

A

Lee Papayanopoulos

*A Handful of Coins …*

Consider deliberately dropping

10penniesand asking your5-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:

- The amount Lulu gets, call it Y, is
a
. That means that the number of "heads" cannot be predicted before that magic toss.*random variable* - Yet, she may state the probability
of any
involving the random variable. For example, the event that she gets exactly 2 cents has probability:*event*

*f*(*2*)* =*
*Prob*(Y*=2*) = (*½*)ış*10*!/(*2*!*8*!)
= *45/1024* = *4.4*%

- Y obeys the
, given by the formula (exclamation marks depict "factorials")*binomial distribution*

*f*(*y*)* =
Prob*(Y*=y*) = (*½*)ış*10*!/(*y*!*10-y*!)

Lulu's winnings can be thought of as the sum Y =

*x*_{1}+... +*x*_{n}. The variables (*x*_{1}...*x*_{n}) 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 afrequencydistribution. To obtain aprobabilityfunction, Lulu divides these values by2^{10}=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

weightedsum Y =5x_{1}+5x_{2}+... +25x_{10}. Its distributionf(y)= Prob(Y=y) is no longerIt is anbinomial., a generalization of the binomial for weighted sums. For this set of coins it has a mean of

omega function72.5and looks like this:

It is very likely that Lulu will walk away with

70or75cents 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 than1/10of1%.If you were to add 3 pennies to your coins, you'd have

as the parameters of1, 1, 1, 5, 5, 5, 10, 10, 10, 25, 25, 25, 25f(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

and18, 7, 3, 9, 5, 6, 21, 30, 27, 15, 61, 12,cents, respectively. You toss them and again Lulu keeps all the "heads."24If the coins are fair, in the conventional sense, do you have any idea what the

of the distribution would be?shape

First, try to imagine and thenclick 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 Apportionmentof 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).