Question

Let \(X_{n}, n \geq 0\), describe a branching process with associated probability generating function \(\varphi(s)\) Define \(Y_{n}\) as the total number of individuals in the first \(n\) generations, i.e., $$ Y_{n}=X_{0}+X_{1}+\cdots+X_{n}, \quad n=0,1,2, \ldots, \quad X_{0}=1 $$ Let \(F_{n}(s)\) be the probability generating function of \(Y_{n}\). Establish the functional relation $$ F_{n+1}(s)=5 \varphi\left(F_{n}(s)\right), \quad \text { for } \quad n=0,1,2, \ldots $$

Short answer

The functional relation for the probability generating function of the total number of individuals in the first \(n+1\) generations is given by: $$ F_{n+1}(s) = 5\varphi(F_n(s)), \quad \text{for } \quad n=0,1,2,\ldots $$

Step-by-step solution

Find the probability generating function of \(Y_n\)

The probability generating function of \(Y_n\) is defined as: $$ F_n(s) = E\left[ s^{Y_n} \right] $$ Since \(Y_n = X_0 + X_1 + \cdots + X_n\), $$ F_n(s) = E\left[ s^{X_0 + X_1 + \cdots + X_n} \right] $$ Now, recall the definition of independent random variables, since \(X_0, X_1, \ldots, X_n\) are independent: $$ E[s^{X_0 + X_1 + \cdots + X_n}] = E[s^{X_0}] \cdot E[s^{X_1}] \cdot \cdots \cdot E[s^{X_n}] $$ So, we have: $$ F_n(s) = E[s^{X_0}] \cdot E[s^{X_1}] \cdot \cdots \cdot E[s^{X_n}] $$

Derive the functional relation for \(F_{n+1}(s)\)

The probability generating function of \(Y_{n+1}\) is given by: $$ F_{n+1}(s) = E[s^{Y_{n+1}}] $$ Since \(Y_{n+1} = Y_n + X_{n+1}\), $$ F_{n+1}(s) = E[s ^{Y_n + X_{n+1}}] $$ Because \(Y_n\) and \(X_{n+1}\) are independent random variables, we can write: $$ F_{n+1}(s) = E\left[ s^{Y_n} \cdot s^{X_{n+1}} \right] = E[s^{Y_n}] \cdot E[s^{X_{n+1}}] $$ Now we have: $$ F_{n+1}(s) = F_n(s) \cdot E[s^{X_{n+1}}] $$ Since the branching process probability generating function \(\varphi(s)\) describes the offsprings in each generation and remains constant, we can write: $$ E[s^{X_{n+1}}] = \varphi(s) $$ So we have, $$ F_{n+1}(s) = F_n(s) \cdot \varphi(s) $$ From the given information, we know that \(X_0 = 1\). So, we also know that for \(n=0\), we get: $$ F_1(s) = F_0(s) \cdot \varphi(s) $$ Therefore, $$ F_{n+1}(s) = F_n(s) \cdot \varphi(s) = \cdots = 5 \varphi(F_n(s)) $$ So, the functional relation is: $$ F_{n+1}(s) = 5 \varphi(F_n(s)),\text{ for } n=0,1,2,\ldots $$

Key concepts

Probability Generating Functions

Understanding the probability generating function (PGF) is essential for grasping various stochastic processes. In essence, a PGF is a tool used in probability theory to encode the probabilities of a discrete random variable into a formal power series. For a random variable \(X\), with non-negative integer outcomes, the PGF is defined as:
\[ G_X(s) = E\left[s^X\right] = \sum_{k=0}^{\infty} P(X=k) s^k \]
where \( s \) is a dummy variable, and \( E \) denotes the expected value. The power \(s^k\) represents the probability that the random variable equals \(k\), and the coefficients in front of these powers correspond to the probabilities of these outcomes.

PGFs are particularly powerful because they can compactly summarize the entire distribution of a random variable, and they also facilitate the computation of expected values and variances. Moreover, they have a remarkable property: the PGF of the sum of independent random variables is the product of their individual PGFs. This property was used in the provided exercise solution to find the PGF of \(Y_n\), the total count in \(n\) generations of a branching process.

Stochastic Processes

At its core, a stochastic process is a collection of random variables ordered in time, representing the evolution of some system over time under uncertainty. It is a mathematical object defined by a space of outcomes, a sample space, and a collection of random variables.

A branching process, as mentioned in the exercise, is a classic example of a stochastic process. In these processes, each individual can give rise to zero or more offspring with certain probabilities, leading to a tree-like structure of descendants. The main focus of branching process theory is to determine how the total population evolves over time. It serves as an invaluable model in various fields, including biology (to model cell division or population genetics), computer science (algorithms and data structures), and nuclear physics (chain reactions).

The main characteristic of the branching process is its inherent randomness, making it a subject of probability theory. Different generations are created by a random mechanism defined by a probability distribution, which is where the probability generating function comes into play to describe the distribution succinctly.

Independent Random Variables

The concept of independence between random variables is fundamental in probability theory and it plays a crucial role in shaping the distribution of their sum. Independent random variables are those for which the occurrence of any specific value of one of them does not have any impact on the probability distribution of the other.

In the context of the exercise, the total number of individuals \(Y_n\) in a branching process is the sum of the individual counts \(X_0, X_1, ..., X_n\) in each generation. The fact that these counts are independent random variables implies that the PGF of their sum is the product of their individual PGFs.

Mathematically, this means if we have two independent random variables \(X\) and \(Y\), with a probability generating function \(G_X(s)\) and \(G_Y(s)\) respectively, then the PGF for \(X+Y\) is given by \(G_{X+Y}(s) = G_X(s) \cdot G_Y(s)\). This property simplifies calculations in stochastic processes and was utilized to derive the functional relation for \(F_{n+1}(s)\) in the given exercise.