Abstract
Wishart and Hirschfeld1 obtained the moment-generating function for the probability distribution of adjacent black-white joins when m points, black or white with probabilities p and q, are arranged at random on a line, by working out the difference equations connecting the moment-generating functions for (m − 1), m and (m + 1) points. Recently, Krishna Iyer2 used this method to darive the moment-generating function for the distribution of black-black joins. The success of the method depends upon the fact that the r-th factorial moment, µ′[r], is equal to r! multiplied by the expectation for obtaining r black-black joins. This property will enable us to work out the difference equations satisfied by the moment-generating functions of a number of distributions arising from m points arranged at random on a line. This note gives in seriatim the difference equations for the following distributions : (i) the number of joins between points of different colours when the points can assume one of k colours with probabilities p1, p2 … pk, (ii) the number of runs of length r of a specified colour, (iii) the number of runs of length r or more of a specified colour, and (iv) the number of triplets, quadruplets, etc., of a specified colour. They are as follow: where ar stands for the monomial symmetric function of degree r in p1, p2, … pk, that is, (∑ p1p2 … pr). where s is the number of points in the s-plet.
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References
Wishart, J., and Hirschfeld, H. O., J. Math. Soc., London, 11, 227 (1936).
Krishna lyer, P. V., J. Ind. Soc. Agric. Stat., 1, 171 (1948).
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KRISHNA IYER, P. Difference Equations of Moment-Generating Functions for some Probability Distributions. Nature 165, 370 (1950). https://doi.org/10.1038/165370a0
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DOI: https://doi.org/10.1038/165370a0
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