The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.