Letter | Published:

Runs in a Sequence of Observations

Naturevolume 168page557 (1951) | Download Citation



IN previous communications1–4 one of us (P. V. K. I.) has discussed a number of distributions arising from m observations belonging to k groups arranged at random on a line. When all the observations are different, m is equal to k, and for this case Kermack and McKendrick5, and Levene and Wolfowitz6 have considered the distribution of the number of runs ‘up’ and ‘down’. A run ‘up’ or ‘down’ is a succession of observations in ascending or descending order. Kendall7 has dealt with the same problem by considering the number of ‘troughs’ and ‘peaks’. When m is greater than k, the sequence of observations will have three kinds of runs, namely, ascending, descending and stationary. This note gives the difference equation satisfied by the probability generating function and the first two moments for the distribution of the total number of junctions for a sequence of m observations with fixed probabilities p1, p2pk, a junction being the meeting point between two runs ‘up’, ‘down’ or stationary. It may be noted that the total number of runs is equal to the total number of junctions increased by unity.

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  1. 1

    Krishna Iyer, P. V., Nature, 160, 714 (1947).

  2. 2

    Krishna Iyer, P. V., Nature, 162, 333 (1948).

  3. 3

    Krishna Iyer, P. V., J. Ind. Soc. Agric. Stat., 1, 173 (1948).

  4. 4

    Krishna Iyer, P. V., Ann. Math. Stat., 21, 198 (1950).

  5. 5

    Kermack, W. O., and McKendrick, M. G., Proc. Roy. Soc., Edin., 47, 228 and 332 (1937).

  6. 6

    Levene, H., and Wolfowitz, Ann. Math. Stat., 13, 58 (1944).

  7. 7

    Kendall, M. G., “The Advanced Theory of Statistics”, 2, 124 (London: Griffin and Co., Ltd., 1946).

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  1. Indian Council of Agricultural Research, New Delhi

    •  & A. S. PRAKASH RAO


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