7.2. Probability Generating Functions, Practical Class 2#
AMSI 2026
Suppose a random integer-valued variable \(X\) is generated as follows:
With probability \(0.6\), \(X\) has probability generating function \(\mu_1(x)\).
With probability \(0.4\), \(X\) has probability generating function \(\mu_2(x)\).
Independently of this choice, let \(Y\) be a Bernoulli\((0.5)\) random variable (so \(Y=0\) or \(1\) with equal probability), and define \(Z = X + Y\).
(a) Find the PGF for \(X\) in terms of \(\mu_1(x)\) and \(\mu_2(x)\).
(b) Find the PGF for \(Y\)
(c) Write down the probability generating function of \(Z\).
(d) Explain how the coefficient of \(x^k\) in this generating function can be interpreted.
Let \(D\) be the number of customers arriving at a store in one hour, with PGF \(\mu_D(x)\). Each customer independently purchases a random number of items, with PGF \(\mu_S(x)\). Let \(T\) be the total number of items purchased in that hour and \(\psi(x)\) be the PGF of \(T\).
(a) Express \(\psi(x)\) in terms of \(\mu_D(x)\) and \(\mu_S(x)\).
(b) Suppose
\[ \mu_D(x) = 0.3 + 0.7x \]and
\[ \mu_S(x) = 0.2 + 0.8x^2. \]Compute \([x^2]\psi(x)\), and explain what this coefficient represents in the context of the problem.
A traffic light allows a random number of cars to pass during each cycle. The PGF for the number of cars passing in a single cycle is \(\mu(x)\). Assume different cycles are independent.
(a) Write down the probability generating function for the total number of cars passing through the light over two consecutive cycles.
(b) Explain how you would use this generating function to find the probability that exactly \(n\) cars pass in two cycles.
(c) Describe how your approach generalises to more cycles.
Consider a Galton-Watson process whose offspring distribution has PGF \(\mu(x)\). Assume that \(X_0=1\), \(X_1=3\), and \(X_2=4\).
(a) Find the PGF for \(X_3\) conditional on the information above in terms of \(\mu(x)\).
(b) The formula you found in (a) should depend on \(X_2\). Does knowing \(X_1\) (or \(X_0\)) provide any useful information for predicting \(X_3\)?
(c) For each of the following distributions, find a formula for \(\mathbb{P}[X_3 = n]\) for \(n=0,1,2,\ldots\). Note that in each case a power of the PGF is a PGF of the same general form, so its series expansion is known:
(i). Poisson: \(\mu(x) = e^{\lambda (x-1)} = \sum_{k=0}^\infty \frac{e^{-\lambda} (\lambda x)^k}{k!}\).
(ii) Binomial: \(\mu(x) = (1-p + px)^N = \sum_{k=0}^N \binom{N}{k} p^k(1-p)^{N-k}x^k\).
(iii) Negative Binomial: \(\mu(x) = \left(\frac{p}{1-(1-p)x}\right)^r = \sum_{N=0}^\infty \binom{k+r-1}{k} p^{r}(1-p)^k x^k\).
(iv) Geometric distribution: \(\mu(x)=\frac{pc}{1-(1-p)x} = \sum_{k=1}^\infty (1-p)^{k-1} p x^k\)? (hint, compare with the Negative Binomial distribution)
(v) (*) Write down an intgral that can be used in a general case where a power of the PGF is not simply another PGF whose Taylor Series is known.
Consider a Galton–Watson branching process with offspring distribution having probability generating function
\[ \mu(x) = p_0 + p_1 x + p_2 x^2 \]where \(p_0 = 0.2\), \(p_1 = 0.4\), \(p_2 = 0.4\).
(a) Let \(\alpha\) denote the probability that the population eventually goes extinct. Write down the equation that \(\alpha\) must satisfy in terms of \(\mu(x)\).
(b) Solve this equation and identify which solution in \([0,1]\) is the correct extinction probability. Explain why any other solutions should be rejected.
(c) Compute the mean number of offspring \(\mu'(1)\) and use it to explain whether extinction is certain or only occurs with some probability less than \(1\).
Consider a continuous-time branching process defined as follows:
Each individual lives for an exponentially distributed amount of time with rate \(r\).
When an individual dies, it is instantaneously replaced by a random number of offspring with probability generating function \(\hat{\mu}(x)\).
All individuals behave independently of one another.
Let \(\Phi(x,t)\) be the probability generating function of the population size \(X(t)\) at time \(t\), starting from a single individual at time \(0\): \(X(0)=1\).
The Backward Kolmogorov equation for this process is
\[ \frac{\partial}{\partial t}\Phi(x,t) = r\bigl(\hat{\mu}(\Phi(x,t)) - \Phi(x,t)\bigr), \]with initial condition \(\Phi(x,0) = x\).
Define
\[ \alpha(t) = \mathbb{P}[\text{the population is extinct by time } t] = \Phi(0,t). \](a) By substituting \(x=0\), derive a differential equation satisfied by \(\alpha(t)\).
(b) Suppose the offspring distribution is Poisson with mean \(2\), so that
\[ \hat{\mu}(x) = e^{2(x-1)}. \]Write the resulting ODE for \(\alpha(t)\) explicitly.
(c) Explain what the limit
\[ \lim_{t \to \infty} \alpha(t) \]represents.
Review the derivation of \(\Phi(x,T_1+T_2)\) in the notes. In the generation-based case we focused on just \(\Phi_{g+1}(x)\). Adapt the derivation of \(\Phi(x,T_1+T_2)\) to find the generation-based equivalent for \(\Phi_{g_1+g_2}(x)\).
We derived the Forward and Backward Kolmogorov Equations for the case of one initial individual. Assume instead that the process begins with \(2\) individuals.
(a) What would need to change about the setup of the Forward Kolmogorov Equation derivation?
(b) What would need to change about the setup of the Backward Kolmogorov Equation derivation?
(c) If you have time, derive the two equations.
Consider the two different definitions of the Continuous-Time Galton-Watson processes. Assume we are using the definition based on individuals in which they have an exponentially distributed lifetime of rate \(r\), corresponding to a Poisson process with rate \(r\). By referring to superposition[1] of Poisson processes, explain why at time \(t_i\) we can assume that the time until the next event is exponentially distributed with rate \(rX(t_i)\).