Question

Consider a Yule process \(\left\\{N_{t}, t \geq 0\right\\}\) with birthrate \(\lambda\) and initial population of size \(1 .\) Find the distribution function of \(N_{t}(x)=\) number of members of the population at time \(t\) of age less than or equal to \(x\).

Short answer

\(N_t(x) = \sum_{k=0}^\infty P(X_{k+1} > x | N_t = k+1)P(N_t = k+1)\) The distribution function of \(N_t(x)\) is found by considering the probabilities of individuals existing at time \(t\) and having age less than or equal to \(x\), where each probability is determined by conditional probability and the cumulative distribution function of the exponential distribution.

Step-by-step solution

Understanding Yule Process

The Yule process, also known as the Yule-Simon process, is a type of branching process where the number of offspring for each member of the population follows a specific distribution, and the birth rate is constant. In this case, our Yule process has a constant birth rate of \(\lambda\) and an initial population of size 1.

Calculate Probability of a Member Having n Offspring

In a Yule process, the probability of a member of the population having exactly \(n\) offspring is given by the geometric distribution with parameter \(\lambda\). Let \(P_n\) denote the probability that a member has exactly \(n\) offspring. This can be calculated as: \(P_n = (1-\lambda)^n\lambda \)

Define Age of Members

Let \(X_i\) be the age of the \(i\)-th member of the population at time \(t\). Given the constant birth rate, the time required for each member to reproduce follows an exponential distribution with parameter \(\lambda\). The age of each member \(i\) at time \(t\) is given by: \(X_i = t - T_i\) where \(T_i\) is the birth time of the individual \(i\).

Determine Number of Members with Age less than or Equal to x

To determine the number of members with age less than or equal to \(x\) at time \(t\), we must find the probability that the age of each member, \(X_i\), is less than or equal to \(x\). We know that \(X_i = t - T_i\), and since the birth time follows an exponential distribution with parameter \(\lambda\), we can use the cumulative distribution function (CDF) of the exponential distribution as follows: \(P(X_i \leq x) = 1 - e^{-\lambda x}\) Thus, the distribution function of \(N_t(x)\), the number of members with age less than or equal to \(x\) at time \(t\), can be determined by considering the probabilities of individuals existing at time \(t\) and having age less than or equal to \(x\).

Derive Distribution Function for N_t(x)

To find the distribution function of \(N_t(x)\), we first consider the partition of the individuals in the population for a given \(k\) offspring, and then sum the probabilities associated for each \(k\). Let \(Q_k\) denote the probability that a member has \(k\) offspring at time \(t\) with age less than or equal to \(x\). Based on conditional probability, we can express \(Q_k\) as: \(Q_k = P(X_{k+1} > x | N_t = k+1)P(N_t = k+1)\) Where \(P(X_{k+1} > x | N_t = k+1)\) represents the probability that the \((k+1)\)-th individual has age greater than \(x\) given that there are \((k+1)\) total individuals at time \(t\), and \(P(N_t = k+1)\) represents the probability that there are \((k+1)\) total individuals at time \(t\). We can then use the probabilities found in Step 2 and Step 3 to find \(Q_k\). After finding \(Q_k\) for each \(k\), we can sum them to find the distribution function for \(N_t(x)\): \(N_t(x) = \sum_{k=0}^\infty Q_k\) The distribution function derived in this step represents the number of members in the population with age less than or equal to \(x\) at time \(t\).