1.11. Properties of Generating functions#
This section is under construction. For now you can ignore it.
As noted previously, we will be interested in the values of
when it contains information about the sequence of coefficients of \(f\).
A few immediate properties can be observed:
\(c_0 = f(0)\)
\(c_0 + c_1 + c_2 + \cdots = f(1)\)
\(c_1 + 2c_2 + \cdots = f'(1)\)
(these only hold if the sums converge - if \(f(x)\) has a singularity between \(0\) and \(1\), the sums do not converge).
1.11.1. Properties of PGFs#
Consider a probability distribution \(p_0, p_1, \ldots\). Let \(f(x)\) be the corresponding PGF. Then
\(f(0)=p_0\)
\(f(1) = 1\)
\(f'(1) = \sum k p_k = \langle K \rangle\) where \(\langle K \rangle\) denotes the average value of \(k\).
\(f''(1) = \sum k(k-1)p_k = \langle K^2-K\rangle\).
\(\left. \frac{d}{dx} \left(x f'(x)\right) \right|_{x=1} = \langle K^2 \rangle\).
These can all be proven by writing out the summations and substituting in \(x=1\).
1.11.2. Self-test#
This self-test focuses on properties of the PGFs
Verify that for a PGF