Sankhya: The Indian Journal of Statistics
One Bulb? Two Bulbs? How Many Bulbs Light Up? -- A Discrete Probability Problem Involving Dermal Patches
C. Radhakrishna Rao, Pennsylvania State University, University Park, USA
M. Bhaskara Rao, University of Cincinnati, Cincinnati, USA
Haimeng Zhang, Concordia College, Moorhead, USA
SUMMARY. A dermal patch is designed to activate some targeted receptors. On Day 1, the patch releases one dose of medicine, which latches onto a receptor and makes it active. On Day 2, the patch releases two doses of medicine, which latch onto two receptors, one dose per receptor. If the receptor is already active, the new dose makes it inactive. If the receptor is inactive, the new dose makes it active. On Day 3, the patch releases three doses of medicine, which latch onto three receptors, one dose per receptor. This continues for ten days with the patch releasing a total of 55 doses progressively. In this paper, we obtain the distribution of the number of receptors active at the end of Day 10.
AMS (2000) subject classification. Primary 60C05, 60B15, 62P10, 62E20.
Key words and phrases. Asymptotic normality, clubbed binomial distribution, de Finetti's theorem, discrete probability, exchangeability, modulo operation, two-element compact group.