Question

(a) A mature individual produces offspring according to the probability* generating function \(f(s) .\) Suppose we have a population of \(k\) immature individuals, each of which grows to maturity with probability \(p\) and then reproduces independently of the other individuals. Find the probability generating function of the number of (immature) individuals at the next generation. (b) Find the probability generating function of the number of mature individuals at the next generation, given that there are \(k\) mature individuals in the parent generation.

Short answer

The probability generating function for the number of immature individuals at the next generation is \((1 - p + ps)^k\), and the probability generating function for the number of mature individuals at the next generation, given that there are \(k\) mature individuals in the parent generation, is \([f(s)]^k\).

Step-by-step solution

Understanding Probability Generating Function

A Probability Generating Function (PGF) is a power series that encodes the probability distribution of a random variable. Given a discrete random variable X with probabilities \(P(X = n) = p_n\), its probability generating function is denoted by \(G_X(s)\) which has the form: \[G_X(s) = \sum_{n=0}^{\infty}p_ns^n\]

Part (a): Probability generating function for immature individuals

We need to find the probability generating function for the number of immature individuals at the next generation. Since an individual matures with probability \(p\), it remains immature with probability \((1-p)\). We can define a new random variable \(Y_i\) for the \(i^{th}\) immature individual offspring in the next generation where \(Y_i = 0\) if the \(i^{th}\) individual matures and \(Y_i = 1\) if it remains immature. Since each immature individual grows to maturity independently, the probability generating function for \(Y_i\) can be given by: \[ G_{Y_i}(s) = P(Y_i=0)s^0 + P(Y_i=1)s^1 = 1 - p + ps \] Now let's look at the total number of immature individuals in the next generation. This is represented by the random variable \(Z = \sum_{i=1}^{k}Y_i\). The probability generating function for \(Z\) is given by the product of the PGFs of all the \(k\) immature individuals as they are all independent. \[ G_Z(s) = \prod_{i=1}^{k}G_{Y_i}(s) = (1 - p + ps)^k \] So, the probability generating function for the number of immature individuals in the next generation is \((1 - p + ps)^k\).

Part (b): Probability generating function for mature individuals

We are given the probability generating function \(f(s)\) for a mature individual producing offspring. Now we need to find the probability generating function for the number of mature individuals at the next generation, given that there are \(k\) mature individuals in the parent generation. Let \(X_i\) denote the offspring of the \(i^{th}\) mature individual. Since a mature individual reproduces according to the probability generating function \(f(s)\), we have: \[ G_{X_i}(s) = f(s) \] Now let's denote by W the total number of mature individuals in the next generation. This is represented by the random variable \(W = \sum_{i=1}^{k}X_i\). The probability generating function for \(W\) is given by the convolution of the PGFs of all the \(k\) mature individuals as they are all independent. \[ G_W(s) = G_{X_1}(s) * G_{X_2}(s) * \cdots * G_{X_k}(s) = [f(s)]^k \] So, the probability generating function for the number of mature individuals at the next generation, given that there are \(k\) mature individuals in the parent generation, is \([f(s)]^k\).

Key concepts

Discrete Random Variables

When we talk about random variables, we are referring to quantities whose values are determined by chance. Discrete random variables, in particular, can take on a finite or countably infinite set of values. These variables are often used in situations where you are counting instances, such as the number of heads in a series of coin flips, the number of students who pass a test, or the number of immature individuals in a population, as discussed in our probability generating function problem.

Each value that a discrete random variable can assume has an associated probability, which must fall between 0 and 1, inclusive. Moreover, the sum of the probabilities for all possible values of the variable must be exactly 1. In the context of the exercise provided, we assigned a variable, say, to the outcome of whether an immature individual remains immature or matures. By defining this variable distinctly, it becomes easier to manage and understand the associated probabilities.

Probability Distribution

The probability distribution of a discrete random variable is a list (or a function) specifying the likelihood of each of its possible outcomes. It gives us a complete description of the randomness of the variable. In the exercise, we're working with a probability generating function (PGF), which is essentially a compact representation of a probability distribution. The PGF encodes all the probabilities associated with a discrete random variable into a single function.

Understanding the PGF is key to solving problems about the next generation's population in our exercise. For each immature individual, the PGF represented the two possible outcomes: remaining immature or maturing. By raising this PGF to the power of the number of individuals (\(k\)), we harnessed the power of probability distributions to efficiently calculate the overall PGF for the population. This is particularly useful when dealing with large or complex distributions because it simplifies calculations and provides a clearer overview of the random variable's behavior.

Independence in Probability

The concept of independence in probability is fundamental to understanding how multiple random processes interact with each other. Two events are considered independent if the occurrence (or non-occurrence) of one event does not change the likelihood of the other event occurring. In the context of our exercise, we assumed that individual organisms reproduce independently.

To illustrate, let's consider two individuals from our population. The chance of one maturing and producing offspring does not affect whether the second does the same, which aligns with the concept of independence. Therefore, when we calculated the total PGF for the number of immature and mature individuals at the next generation, we took the product of each individual's PGF. This mathematical maneuver is valid only because of the independence among individuals. Had there been any interdependencies, a different approach would be needed to account for those relationships in our probability models.