2.1 The Probability Generating Function
Definition 2.1.1. Let \(X\) be a discrete random variable taking values in \(\{0,1,2,\dots \}\). Its probability generating function is \[G_X(t) = E\left (t^{X}\right ) = \sum _{x=0}^{\infty } t^{x}\,P(X=x),\] defined for every real \(t\) for which the series converges.
The series always converges for \(\left |t\right |\leq 1\), because the probabilities sum to one; how much further it converges depends on the distribution.
Note. The name is literal. The coefficient of \(t^{x}\) in the power series is \(P(X=x)\), so the function generates the probabilities, and reading them off is a matter of expanding the series.
Example 2.1.2. A discrete random variable \(X\) has probability function \[P(X=x) = \frac {x}{15},\qquad x = 1,2,3,4,5 .\] Derive a formula for \(G_X(t)\).
Solution. Substituting directly into the definition, \[G_X(t) = \sum _{x=1}^{5} t^{x}\,\frac {x}{15} = \frac {1}{15}t + \frac {2}{15}t^{2} + \frac {3}{15}t^{3} + \frac {4}{15}t^{4} + \frac {5}{15}t^{5}.\] The coefficients are the probabilities, as the definition promises, and they sum to \(15/15 = 1\) — which is the check \(G_X(1)=1\) below.
Recovering the probabilities
Writing the series out, \[G_X(t) = t^{0}P(X=0) + t^{1}P(X=1) + t^{2}P(X=2) + t^{3}P(X=3) + t^{4}P(X=4) + \cdots \] Differentiating and setting \(t=0\) kills every term but one at each stage: \[G_X'(t) = P(X=1) + 2t\,P(X=2) + 3t^{2}P(X=3) + \cdots \quad \Longrightarrow \quad G_X'(0) = P(X=1),\] \[G_X''(t) = 2P(X=2) + 6t\,P(X=3) + \cdots \quad \Longrightarrow \quad G_X''(0) = 2\,P(X=2),\] and in general \[P(X=n) = \frac {1}{n!}\,G_X^{(n)}(0),\qquad n = 0,1,2,\dots \]
Example 2.1.3. A random variable \(X\) has \(G_X(t) = 0.3 + 0.2t^{2} + 0.5t^{3}\). Write down its probability distribution.
Solution. Insert the missing powers explicitly: \[G_X(t) = 0.3\,t^{0} + 0\,t^{1} + 0.2\,t^{2} + 0.5\,t^{3},\] and read off the coefficients. \[\begin {array}{c|cccc} x & 0 & 1 & 2 & 3\\ \hline P(X=x) & 0.3 & 0 & 0.2 & 0.5 \end {array}\] No differentiation is needed when the generating function is already a polynomial; the coefficients are the answer.
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