Question

Waiting time distributions One informative way of analyzing the frequency of events is illustrated in Figure \(18.47(\mathrm{C})\). Here, the cumulative probability of events is obtained by measuring how many events have occurred if we wait a time \(t .\) This cumulative probability is fit to the equation \(n=N\left[1-\mathrm{e}^{-t / \tau}\right],\) where \(N\) is the total number of events. Show that his functional form is precisely what is expected for a process characterized by the waiting time distribution \(p(t)=\mathrm{e}^{-t / \tau}\)

Short answer

The cumulative probability is precisely what is expected for a process characterized by the waiting time distribution \(p(t)=\mathrm{e}^{-t / \tau}\) once we define the CDF as the integral of the PDF and solve the integral, which results in the original cumulative probability as given in the problem statement.

Step-by-step solution

Define distributions

Define the cumulative distribution function (CDF) as \(N\left[1-\mathrm{e}^{-t / \tau}\right]\) and the probability density function (PDF) of the waiting time distribution as \(p(t)=\mathrm{e}^{-t / \tau}\).

Relate the distributions

The CDF is defined as the integral from negative infinity to \(t\) of the PDF. Hence, for the waiting time \(t\), it can be expressed as \(N(1 - \int_{-\infty}^{t} \mathrm{e}^{-t / \tau} dt)\).

Simplify the integral

Solving the integral \(\int_{-\infty}^{t} \mathrm{e}^{-t / \tau}dt = -\tau \left[ \mathrm{e}^{-t / \tau} \right]_{-\infty}^{t}\). The lower limit gives zero and the upper limit gives \(-\tau \mathrm{e}^{-t / \tau}\). Hence, the integral can be simplified to be \(\tau \mathrm{e}^{-t / \tau}\).

Put it all together

Substitute the result of the integral into the expression for the cumulative distribution function. It results as \(N(1 - \tau \mathrm{e}^{-t / \tau})\). This is exactly what was wanted to be shown, since this matches the original cumulative probability given in the statement.

Key concepts

Waiting Time Distributions

Waiting time distributions are incredibly useful in statistics and probability for modeling the time between successive events. The key idea is to examine how long one has to wait until an event occurs, with applications ranging from customer service to natural phenomena.

Typically, the 'event' can be anything like receiving an email, a meteorite hitting the earth, or a customer arriving at a store. The waiting time, represented by a random variable, could have various probability distributions depending on the nature of the events. One of the common models is the exponential distribution, characterized by a constant rate of occurrence.

Key Properties of Waiting Time Distributions

• Memorylessness: The distribution has the property that the probability of an event occurring in the next moment is independent of how much time has already passed.
• Rate of Occurrence: The rate at which events occur is consistent, meaning the average waiting time remains stable over time.
• Continuous Probability Distribution: Waiting time is often modeled using continuous probability distributions since time can be measured to any degree of granularity.

Probability Density Function (PDF)

The probability density function (PDF) is a statistical expression that describes the likelihood of a continuous random variable taking on a particular value. Unlike discrete random variables, continuous variables require a density function because the probability of the variable taking on an exact value is essentially zero since there are infinitely many possible values.

The PDF is integral to understanding how values of a random variable are distributed across a range. It helps identify where the random variable is more likely to fall, which is visualized as areas under the PDF curve.

Applications of PDF

• Assessing Probabilities: The area under the PDF curve between two points corresponds to the probability of the random variable falling within that range.
• Analyzing Distributions: Different functions represent different types of distributions, each with specific applications and implications, such as the exponential function for modeling waiting times.
• Statistical Analysis: PDFs are crucial for statistical tools, hypothesis testing, and in various fields such as finance, engineering, and environmental studies.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is another fundamental concept in the study of probability and statistics. It provides the cumulative probability associated with a given value of a random variable. Essentially, the CDF indicates the probability that a random variable is less than or equal to a certain value.

In the context of waiting time distributions, the CDF represents the total probability of an event occurring within a given timeframe. It is calculated as the integral of the PDF from the lower limit of the random variable up to a point of interest.

Characteristics of the CDF

• Monotonic: The CDF is always a non-decreasing function as you move from left to right along the horizontal axis.
• Range Bound: The CDF ranges from 0 to 1, indicating zero probability at the beginning and certainty at the far right end.
• Provides Insight: It allows for direct calculation of the probability of an event occurring within a specified interval, which is invaluable in practical applications.

By understanding both the PDF and CDF, students can gain a deeper insight into how random variables behave within a given distribution and determine the likelihood of events over intervals, which is fundamental in fields such as risk assessment, quality control, and policy making.