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Branching Problems and PGFs

name: inverse class: center, middle, inverse

Branching Problems and PGFs

The Probability Generating Function

$X$ is a discrete random variable with support $\mathbb{Z}_{\geq 0}$ and probability mass function $p_X$. Then we define the probability generating function $G_X(🙂)$ as

$$G_X(🙂)\equiv E[🙂^X] = \sum_{x=0}^\infty \left[ p_X (x) \cdot 🙂^x \right]$$

Useful Properties of the PGF

We can easily sum variables:

If $Y$ and $W$ are independent (discrete random variables with support $\mathbb{Z}_{\geq 0}$), and $X\sim aY + bW$, then

$$G_X(😬)=G_Y(😬^a)\cdot G_W(😬^b)$$

Useful Properties of the PGF

We can easily sum variables.

  • In particular, if $x_1,X_2,...,X_N$ are iid, and $Y\sim \sum_i X_i$, then

$$G_Y(🙂) = [G_X(🙂)]^N$$

Useful Properties of the PGF

Compound Distributions

If $N$ itself is a discrete random variables with support $\mathbb{Z}_{\geq 0}$, which is independent of $X_i$, and if $Y\sim \sum^N_{i=1} X_i$ is the sum of $N$ iid draws of $X$, then

$$G_Y(🙂) = G_N(G_X(🙂))$$

Useful Properties of the PGF

Compound Distributions

If $N$ itself is a discrete random variables with support $\mathbb{Z}_{\geq 0}$, which is independent of $X_i$, and if $Y\sim \sum^N_{i=1} X_i$ is the sum of $N$ iid draws of $X$, then

$G_Y(🙂) = G_N(G_X(🙂))$

In particular, if $N$ is iid to $X$, then

$$G_Y(🙂) = G_X(G_X(🙂))$$

Useful Properties of the PGF

Moments

Derivative wrt 🙂 is $\sum_{x=0}^\infty \left[x \cdot p_X (x) \cdot 🙂^{x-1} \right]$ and therefore

$$G^\prime(1)=E[X]$$

Useful Properties of the PGF

Moments

Derivative wrt 🙂 is $\sum_{x=0}^\infty \left[x \cdot p_X (x) \cdot 🙂^{x-1} \right]$ and therefore

$$G^\prime(1)=E[X]$$

And in general, the $n$th derivative gives the $n$th factorial moment.

$$G^{\prime n} (1) = E\left[\frac{x^!}{(x-r)!}\right]$$

Useful Properties of the PGF

Zeros:

$$G_X(0) = \sum_{x=0}^\infty \left[ p_X (x) \cdot 0^x \right] = p_0$$

Useful Properties of the PGF

Note that if $X$ is a constant random variable equal to 1, then

$$G_x(🙂) = \sum_{x=0}^\infty \left[ p_X (x) \cdot 🙂^x \right]$$

$$G_x(🙂) = 🙂$$

This also means that the pgf for $1+Y$ is $🙂\cdot G_Y(🙂)$

Useful Properties of the PGF

Also we can get the pmf back out

$$p_X(x) = \frac{1}{k!}\frac{\partial^k}{\partial x^k} G_0 |_{x=0}$$

Applications to Branching Problems

  • Start with a single individual.
  • This individual has offspring distribution $Z$
  • All of their descendents also have iid offspring distributions $Z$

Applications to Branching Problems

If $G=G_Z$, and the 0th generation has 1 individual, then

  • The number of descendents in the first generation has generating function $G(🙂)$
  • '' in the second generation '' $G(G(🙂)) = G^2(🙂)$
  • '' in the third generation '' $G(G((🙂))) = G^3(🙂)$
  • '' in the $g$th generation is $G^g(🙂)$

Applications to Branching Problems

Extinction

  • $G(0)$ gives the chance that the initial individual will have no children
  • $G(G(0))$ gives the chance they will have no grandchildren
  • $G^g(0)$ gives the chance that the lineage will have died out by generation $g$.

Applications to Branching Problems

Extinction

The chance that the lineage will ultimately go extinct is

$$\lim_{g\to\infty} G^g(0) = G^\infty(0)$$

Applications to Branching Problems

Extinction

The chance that the lineage will ultimately go extinct is

$$\lim_{g\to\infty} G^g(0) = G^\infty(0)$$

  • Note that $G^\infty$ has the property that $G^\infty(0)=G(G^\infty(0))$,
  • and that there is at most one $💀\in (0,1)$ such that $G(💀)=💀$
  • Iff $\mu_Z = G^\prime(1)\geq 1$, then there is a unique positive solution for $G^\infty(0)$ $\in(0,1)$
  • If $\mu_X < 1$, then $G^\infty(0)=1$

Example Paper 1:

Superspreading and the effect of individual variation on disease emergence

2005 Nov - J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp & W. M. Getz

Superspreading and the effect of individual variation on disease emergence

  • Each individual $x$ is assumed to spread disease according to a Poisson process with individual specific mean $v_x$.
  • These individual attack rates are Gamma-distributed
  • The Gamma-Poisson mixture distribution has pgf

$G_Z(🙂) = \left(\left(1-🙂\right)\frac{R_{0}}{k}+1\right)^{-k}$

  • High dispersion parameter $k$ makes outbreaks more volatile. Has implications for the targetting of government intervention

Example Paper 2:

Spread of epidemic disease on networks

M. E. J. Newman

Spread of epidemic disease on networks

  • The degree distribution of nodes has mean $z$ and pgf labelled $G_0$

  • Choose a random edge, and travel to one of it's endpoints. The degree distribution of such nodes is labelled $G_1$ and $G_1(x)=\frac{G^\prime(0)}{z}$.

  • Edge percolation: edges are removed iid with probability $1-T$

    • Degree distribution of non-removed edges is labelled $G_0(x;T)$
    • Degree distribution of non-removed edges of nodes reachable by travelling along a random non-remove edge is labelled $G_1(x;T)$

  • Relationships are establish between these distributions.