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Let X be a Poisson variable with parameter λ with a probability mass function, f(k), where k = 0, 1, 2 … We know the index of log-concavity is the function rf(k) = f(k)^2/(f(k-1)f(k+1) = (k+1)/k>1. So, Poisson is log-concave.

Many books have: "The random variable Y is dispersive if, and only if, Y has a logconcave density."

Poisson variable is discrete. Do we have: A Poisson variable X is dispersive?

Simon
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  • Dispersive order: Let X and Y be random variables with quantile functions F-1 and G-1 respectively. If F-1(b)- F-1(a) <= G-1(b)- G-1(a) whenever 0 <a<=b < 1; then X is said to be smaller than Y in the dispersive order. – Simon Oct 02 '19 at 04:04
  • Or, more specifically, let X be a Poisson variable with parameter λ1 with a probability mass function. Let Y be a Poisson variable with parameter λ2 with a probability mass function. Let λ2 > λ1. Is Y more dispersive than X? – Simon Oct 04 '19 at 16:58

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