Question

Determine the distribution of the total life \(\beta_{t}\) of the Poisson process.

Short answer

The total life of the Poisson process, \(\beta_{t}\), follows a gamma distribution. The PDF of \(\beta_{t}\) is given by \(f(t; \lambda, n) = \frac{e^{-\lambda t}(\lambda t)^{n - 1}}{(n-1)!}\), for \(t > 0, \lambda > 0, n = 1, 2, ... \). The parameters of the gamma distribution are \(n\) and \(\lambda\), where \(n\) is the shape parameter and \(\lambda\) is the rate or inverse scale parameter.

Step-by-step solution

Understand the concept of Poisson process

A Poisson process is a stochastic process whereby events occur continuously and independently at a constant average rate. We often use it to model events such as the number of phone calls received by a call center per hour. In this case, for n events in a Poisson process, the time until the nth event is called the total life of the process. This can be denoted as \(\beta_{t}\).

Identify distribution of \(\beta_{t}\)

The total life \(\beta_{t}\) of a Poisson process follows a gamma distribution. This waiting time concept is inherently related to the Poisson process. The probability density function (PDF) of a Poisson process is defined as: \(f(t; \lambda, n) = \frac{e^{-\lambda t}(\lambda t)^{n - 1}}{(n-1)!}\), for \(t > 0, \lambda > 0, n = 1, 2, ... \) Where, \(f(t; \lambda, n) \)represents the PDF of time until the nth event, \(\lambda\) is the rate at which the events occur and \(n\) is the nth event.

Recognize parameters of the gamma distribution

A randomly selected Poisson event from the distribution represents the time till the nth event occurs, given by \(\beta_{t}\). Here \(n\) is shape parameter and \(\lambda\) is the rate or inverse scale parameter of the gamma distribution.

Key concepts

Stochastic Processes

When we talk about a stochastic process, we're referring to a mathematical model that describes a sequence of events happening in a space and evolving over time in a random manner. These processes are essential in various fields including finance, queueing theory, and physics. A Poisson process, which we focus on in this context, is a classic example of a stochastic process. It assumes that events occur independently, and are spaced out randomly over time at a constant mean rate. Understanding the fundamental idea that events in this process do not influence each other and the timing of one event does not impact the timing of the next is critical in grasping the underpinnings of a Poisson process. This concept sets the stage for you to dive deeper into how these random phenomena are mathematically formulated and examined.

Because events in a Poisson process occur with a known constant rate, this process can model occurrences such as natural phenomena, machine breakdowns in factories, or network traffic amongst several applications.

Gamma Distribution

Now, let's explore the gamma distribution, which comes into play when considering the 'waiting time' until a certain number of events happen in a Poisson process. The gamma distribution is a two-parameter family of continuous probability distributions. It's exceptionally useful as it generalises other distributions such as exponential and chi-squared distributions. In the context of the Poisson process, the gamma distribution is used to model the time elapsed until a predetermined number of events have occurred.

For instance, if we're examining lifetimes of certain components or the waiting time for the next bus to arrive given a certain schedule, the gamma distribution provides the framework we need for statistical analysis. It's crucial to understand that the shape of the gamma distribution can significantly vary depending on its parameters; hence it can range from being exponentially shaped to more bell-curved forms.

Probability Density Function

Moving to the next cornerstone concept, the probability density function (PDF) is a function that describes the likelihood of a continuous random variable to take on a particular value. The PDF is integral to understanding how probabilities are distributed across different outcomes for stochastic processes. Specifically, for the gamma distribution associated with a Poisson process, the PDF gives us the likelihood of the time until the nth event occurs.

In the formula provided earlier, the PDF for the gamma distribution is expressed mathematically, representing the probability of observing the waiting time 't' until the nth event happens. It is based on two parameters: the rate \(\lambda\) at which the events occur and the sequence number of the event \(n\). Taking time to comprehend the PDF will help you make predictions about the timing of future events in a Poisson process.

Poisson Distribution Parameters

Finally, let's focus on the parameters of the Poisson distribution, which heavily influence the characteristics of the stochastic process we're examining. The two main parameters are the rate of occurrence \(\lambda\) and the event count \(n\). The rate \(\lambda\) represents the average number of events in a given time frame, and in the case of our total life concept \(\beta_t\), it determines the mean waiting time between consecutive events. On the other hand, \(n\), the event count, specifies the occurrence of the nth event.

Understanding \(\lambda\) - The Rate Parameter

In a Poisson distribution, a higher \(\lambda\) means events happen more frequently. It is intuitively understood as how busy the system is. For instance, in a call center scenario, \(\lambda\) could represent the average number of calls per hour.

Recognizing \(n\) - The Shape Parameter

The other parameter, \(n\), influences the shape of the gamma distribution. When \(n\)=1, the gamma distribution simplifies to an exponential distribution, indicating that we are looking at the time until the first occurrence. As \(n\) increases, the distribution's shape becomes more complex and requires careful analysis for interpretation.

Mastery of these parameters and their interplay is critical for harnessing the full predictive power of the Poisson process. Knowing how to manipulate and interpret \(\lambda\) and \(n\) helps in various applications, ranging from risk assessment to managing business operations. By thoroughly understanding these parameters, you will be able to apply the Poisson distribution effectively in solving real-world problems.