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Given a polynomial $P(x) = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n$ with non-negative coefficients, is there a standard name for (the function of $p_1,...,p_n$ equal to) the variance of an integer-valued random variable that takes the value $k$ with probability $p_k / P(1)$ (for $0 \leq k \leq n$) ?

This quantity can be expressed in terms of the values of $P$ and its derivatives at 1 (though that's not relevant for my purposes).

In a course I'm teaching, I am inclined to simply call this quantity the "variance" of $P(x)$ (or maybe the "exponent variance", to be slightly clearer), but if there's a standard term I'll use that instead.

Likewise, I am inclined to refer to $P(x)/P(1)$ as a "stochastic polynomial" (i.e., a polynomial with non-negative coefficients summing to 1), in analogy with the term "stochastic matrix" (despite the possibility that some people may think I mean a random polynomial). Again, if there's a comparably concise term that's already in use, I won't invent new terminology.

Lastly, when $P(1)=1$, $P(x)$ is called the "probability generating function" of the associated probability mass function; is there standard terminology going in the reverse direction (from the polynomial to the probability distribution)?

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I only remember hearing it called the random variable (or distribution) with pgf $P(x)/P(1)$. –  Brendan McKay Feb 23 '12 at 10:21
    
Maybe you could call P(x) a "measure generating function" instead of a "probability generating function" when the coefficients of $P(x)$ are non-negative but $P(1) \neq 1$. –  Jon Peterson Feb 24 '12 at 12:08

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