In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:

$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{z+1}{2}$$

Author defines gf of a half-coin (a coin with sides $n=0,1,2,...$ with some sides having negative probability):

$$G_{X_{0.5}}(z) = \sqrt{\frac{z+1}{2}} = \frac{1}{\sqrt{2}}\sum_{k=0}^{\infty} \binom{1/2}{k} z^k$$

That is, $$P(X = k) = \frac{1}{\sqrt{2}} \binom{1/2}{k}$$

If we flip two independent half-coins, their sum is 0 or 1 with probability 1/2. So it's like flipping one fair coin.

This is based on pgf of sums of independent random variables being the products of the pgfs of the random variables.

Later on, there's this fundamental theorem saying that for any gf $f$, there exists two pgfs $g, h$ such that

$$fg = h$$

How is that related to the half-coin?

What are the $g$ and $h$ that correspond to $$f(z) = \sqrt{\frac{z+1}{2}}$$


I was expecting something like two gfs for one pgf as was done with the fair coin and the two half coins (product of two gfs for the half coins is the pgf for the fair coin).


1 Answer 1


How is that related to the half-coin?

It provides intuitive interpretation for the half-coin and any other generalized (with negative probabilities) random variables, the way to think about them. This interpretation is given in the paragraph after the fundamental theorem. Namely, if $\mathcal{X}$ is a generalized random variable (with pgf $f$), there exist two ``classical'' independent random variables $Y$ and $Z$ (with pgfs $g$ anf $h$) such that $\mathcal{X}+Y=Z$ in distribution.

What are the $g$ and $h$ that correspond to $f(z)=\sqrt{(z+1)/2)}$?

I don't think the answer is simple and nice-looking. Otherwise author would give it. The fundamental theorem just says that there exists $g(x)=\sum_{n=0}^\infty p_nz^n$ and $h(x)=\sum_{n=0}^\infty q_nz^n$ such that $p_n\geq0$, $q_n\geq0$, and $f(z)g(x)=h(z)$.

  • 1
    $\begingroup$ I don't understand. What's the relation with the half-coin? The half-coin is the script X so what are the Y and Z? The fundamental theorem looks to have come out of nowhere. No proof given either? $\endgroup$
    – BCLC
    Commented Jul 2, 2016 at 9:30
  • 2
    $\begingroup$ Y and Z are classical random variables with pgfs g and h. The fundamental theotem guaranties existence of such Y and Z, but does not provide explicit formulas to find them. The paper does not provide proof, but it cites the original paper that contains proof. Does it make sense now? Or my explanation is vague? $\endgroup$
    – Kostia
    Commented Jul 2, 2016 at 18:02
  • $\begingroup$ OH THANKS. How dumb of me. I thought not so long ago meant earlier in the papet $\endgroup$
    – BCLC
    Commented Jul 2, 2016 at 19:03

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