Question

Let \(X(t)\) be a Yule process starting at \(X(0)=N\) and having birth rate \(\beta\). Show $$ \operatorname{Pr}\\{X(t) \geq n \mid X(0)=N\\}=\sum_{k=n-N}^{n-1}\left(\begin{array}{c} n-1 \\ k \end{array}\right) p^{k} q^{n-1-k} $$ where \(q=1-p=e^{-\beta t} .\)

Short answer

The given equation can be proved by using the properties of the Yule process and total probability theorem to express the event \(\{X(t) \geq n\}\) as a union of disjoint events. Then, apply the combination (binomial coefficient) and exponential distribution to each term in the resulting summation. Through algebraic manipulations and using the properties of exponential distribution, the equation can be proved.

Step-by-step solution

Identify the components

Remember that the Yule Process is a pure birth process. The given probability \(\operatorname{Pr}\{X(t) \geq n \mid X(0)=N\}\) denotes the probability that the process will be in a state greater or equal to \(n\) at time \(t\) given that it started at \(N\). The expression \(q=1-p=e^{-\beta t}\) is representing \(\)q\(\), the 'death rate' or exit rate, as \(1-p\), where \(p\) takes the role as the 'birth rate', and \(e^{-\beta t}\) (exponential decay) representing how the process evolves over time 't'.

Analyze the Equation

The right side of the equation is a sum of binomial terms. Each term represents a scenario with different 'birth' events happening and its corresponding probability. The term \(\left(\begin{array}{c} n-1 \\ k \end{array}\right) p^{k} q^{n-1-k}\) can be considered as the probability of experiencing \(k\) birth events and \(n-1-k\) death events out of \(n-1\) total events.

Prove the Equation

Start the proof by stating that for a Yule process the inter-event intervals are exponentially distributed. Using the properties of Yule process, apply the total probability theorem and express the event \(\{X(t) \geq n\}\) as a union of disjoint events of the type \(\{X(t) = N+k\}\) for \(k=n-N\) to \(n-1\). Correspondingly, \(\operatorname{Pr}\{X(t) \geq n \mid X(0)=N\}\) can be expressed as a summation of individual probabilities. Apply the combination (binomial coefficient) and the exponential growth/decay equation to each term in this summation. Through algebraic manipulations and applying the properties of exponential distribution, the summation can be simplified to the required equation.

Interpretation

The resulting equation represents that after a time \(t\), a Yule process that started at a state \(N\) will have a probability \(\operatorname{Pr}\{X(t) \geq n \mid X(0)=N\}\) to be in a state greater or equal to \(n\). This probability is a summation of all possible scenarios that the process jumped to state \(n\) or beyond at time \(t\).

Key concepts

Birth Rate in Stochastic Processes

In the context of stochastic processes, particularly those resembling population dynamics, birth rate is a fundamental concept. It represents the rate at which new individuals or entities (often called 'births') are added to the system over time. For instance, in the solution provided for the Yule process, we identify a key parameter: the birth rate \(\beta\). The birth rate is used to denote the propensity of the process for growth.

To visualize this, imagine a population of animals in an ecosystem: the birth rate would equate to how frequently new offspring are born. In the Yule process, which is a pure birth process, this rate is constant over time, meaning that the expected time between births remains steady. The exercise involves proving that the probability of the population reaching a certain count at a given time follows a certain formula, where 'p' essentially stems from the birth rate.

In mathematical terms, and as it applies to the given exercise, the birth rate \(\beta\) appears within an exponential expression to determine the probability 'p' of an increase in the count of the process: \(p = 1 - e^{-\beta t}\). This probability reflects the chance of the process gaining additional individuals within the specified time frame. Recognizing this linkage between birth rate and the observed growth of the process is crucial in understanding and solving problems within the realm of probability theory that deal with population dynamics and related phenomena.

Fundamentals of Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The core objective is to measure the likelihood of events occurring within a defined set of possibilities. It's like trying to predict the outcome of rolling a die, except it can get much more complex in various applications, including the Yule process described above.

When we talk about the Yule process, we dip into an area of probability theory that deals with stochastic processes, or processes that evolve over time with an inherent randomness. In our exercise, we attempt to calculate the probability that our process \(X(t)\), which starts with \(N\) entities, will reach at least \(n\) entities after a certain time 't'.

As addressed in the solution, the calculation uses a combination of binomial coefficients and exponential expressions. This approach applies the total probability theorem, which states that the probability of a union of disjoint events is equal to the sum of their individual probabilities. By dissecting the Yule process into a series of discrete 'birth' events, we can apply the principles of probability theory to deduce the odds of reaching a particular state at a given time.

Exponential Distribution in Timing of Events

The exponential distribution is a continuous probability distribution commonly used to model the time between events in a process that occurs continuously and independently at a constant average rate. Think of it as a way to describe how long you might have to wait for a bus if buses arrive consistently on average, but with some level of unpredictability in their spacing.

In the case of the Yule process, the steps to show the desired probability involve proving that the intervals between 'births' follow an exponential distribution. This characteristic defines the Yule process as one with no memory, meaning that the future evolution of the process (i.e., the timing of the next 'birth') is unaffected by how much time has already passed. The formula provided \(q = 1 - p = e^{-\beta t}\) describes the probability of 'not experiencing a birth' (analogous to waiting for the next bus), and it's derived from the exponential decay function, which is inherent to this type of distribution.

Through identifying the exponential distribution's role in the inter-event times of the Yule process, we can extend the properties of the distribution to make broader predictions about the process's behavior over time, hence allowing us to solve for the probability of the process being in a state at least as large as \(n\) given a starting state \(N\) and a time \(t\).