Package jline.api.moment

Class Moment_binomial_from_tail

java.lang.Object
jline.api.moment.Moment_binomial_from_tail
public final class Moment_binomial_from_tail extends Object
Binomial moments from survival (tail) probabilities.

For a nonnegative integer random variable N with survival sequence t_m = P(N >= m), 

   b_j = E[nchoosek(N,j)] = sum_{m>=j} nchoosek(m-1,j-1) * t_m,  j >= 1

with b_0 = t_0 = 1. Unlike every other edge of the house of moments this transform is UPPER triangular, so it consumes the whole tail: the result is exact only if the sequence covers the support, that is t_m = 0 beyond the last element supplied. This is the natural entry point for a closed queueing network, whose queue lengths are bounded by the population and whose joint survival probabilities are ratios of normalizing constants.

Truncating the tail early yields a strict LOWER bound on every b_j, since all the coefficients and all the tail values are nonnegative. The bound is not inherited by the central moments downstream, whose conversion alternates in sign.

Reference: A. Heindl and A. van de Liefvoort. Moment conversions for discrete distributions. PMCCS, 2003.

Since:
LINE 3.0

  • Method Details

    • moment_binomial_from_tail

      public static Matrix moment_binomial_from_tail(Matrix t)
      Converts survival probabilities into binomial moments.
      Parameters:
      t - column vector of length n+1 holding t_0,...,t_n, element m being P(N >= m) and element 0 being 1
      Returns:
      column vector of length n+1 holding b_0,...,b_n