Problem 20 - Calculus for Biology and Medicine

Question

20\. Earthquake Frequency The time between two earthquakes is sometimes modeled using a Poisson process model. According to this model the probability that a second earthquake follows within a time \(t\) of the first earthquake is: $$ P(t)=1-e^{-\lambda t} $$ where \(\lambda\) is a positive constant. We define the average time between earthquakes to be a time \(T\) as follows: The probability that a second earthquake follows the first earthquake within time \(T\) is exactly \(1 / 2\), i.e.: $$ 1 / 2=1-e^{-\lambda T} $$ \(T\) is a function of \(\lambda\). Show that, if the coefficient \(\lambda\) is increased in the model, then the average time between earthquakes will decrease.

Step-by-Step Solution

Answer

Increasing \( \lambda \) decreases the average time \( T \) between earthquakes.

Step 1: Understanding the Given Equation

We begin with the provided equation \( \frac{1}{2} = 1 - e^{-\lambda T} \). This equation states that the probability of a second earthquake occurring within time \( T \) is \( \frac{1}{2} \). Our goal is to analyze how changes in \( \lambda \) affect \( T \).

Step 2: Rearranging the Equation for Analysis

To isolate \( T \), we first solve the equation for \( e^{-\lambda T} \):\[ \frac{1}{2} = 1 - e^{-\lambda T} \] Subtract 1 from both sides \[ -\frac{1}{2} = -e^{-\lambda T} \] Multiply both sides by \(-1\): \[ \frac{1}{2} = e^{-\lambda T} \].

Step 3: Solving for \( T \)

By taking the natural logarithm of both sides, we get \( \ln\left(\frac{1}{2}\right) = -\lambda T \). Now, solve for \( T \): \[ T = -\frac{\ln\left(\frac{1}{2}\right)}{\lambda} \].

Step 4: Analyzing the Expression

The average time \( T \) is given by \( T = \frac{\ln(2)}{\lambda} \) after recognizing that \( -\ln\left(\frac{1}{2}\right) = \ln(2) \). This equation reveals that \( T \) is inversely proportional to \( \lambda \).

Step 5: Evaluating the Effect of Changing \( \lambda \)

Since \( T = \frac{\ln(2)}{\lambda} \), if \( \lambda \) increases, the denominator of the expression becomes larger, causing \( T \) to decrease. Thus, an increase in \( \lambda \) results in a decrease in the average time between earthquakes.

Key Concepts

Poisson Process Model

The Poisson process model is a statistical method used to describe the occurrence of random events over a fixed period of time or space. In our context, it models the frequency of earthquakes. This model assumes that earthquakes occur independently, meaning one earthquake does not influence the timing of the next. These events happen at a constant average rate, which is denoted by the symbol \(\lambda\).

The Poisson process is particularly useful in situations where we need to predict the probability of a number of events happening within a given timeframe. For earthquakes, it helps to understand not just the likelihood of an event occurring, but also patterns over longer periods.

When applying the Poisson process to earthquakes, we're often interested in the time between these events. This time is not fixed, and the model helps to calculate this using exponential functions, which we will discuss later.

Probability

In the context of earthquake frequency, probability measures how likely it is for the next earthquake to occur within a specific timeframe after the first one. The probability function given as \(P(t)=1-e^{-\lambda t}\) helps determine this likelihood within time \(t\).

This formula reflects the density of the time intervals between earthquakes. Here, \(1-e^{-\lambda t}\) represents the cumulative distribution function of the exponential distribution. The closer this probability is to 1, the more likely it is that you will experience a second earthquake within that time frame.

The given model sets a scenario where you're 50% likely to have a second earthquake within time \(T\). Through manipulating the probability model, it’s possible to establish relationships between time intervals, dictated by the constant \(\lambda\).

Exponential Function

The exponential function, particularly \(e^{-\lambda t}\), is crucial in modeling earthquake frequency through the Poisson process. It describes the probability of the time between earthquakes exceeding \(t\) with \(\lambda\) as the rate parameter.

This exponential decay captures the essence of how rapidly the probability of the occurrence of another earthquake decreases as time progresses without one happening.

• \(e^{-\lambda t}\) quickly approaches zero as \(t\) increases, indicating a decreasing likelihood of going for a longer time without an earthquake.
• At \(t=0\), the value is 1, suggesting 100% certainty at the start.

Its role is vital, as it connects the physical expectations of events with mathematical predictions, showing how quickly probabilities change over time as governed by the average rate \(\lambda\).

Average Time Calculation

To calculate the average time between earthquakes, we use the formula \(T = \frac{\ln(2)}{\lambda}\). This is derived from the probability equation set to \(\frac{1}{2}\), representing a 50% chance that a second earthquake occurs within time \(T\).

The average time \(T\) is inversely proportional to \(\lambda\), the rate parameter.

• As \(\lambda\) increases, the denominator of the expression becomes larger.
• This causes the overall value of \(T\) to decrease, indicating shorter times between earthquakes.

This inverse relationship emphasizes that in areas with a high rate of earthquake occurrences (large \(\lambda\)), the average time between earthquakes will be relatively short. It aligns logically with the understanding that more frequent earthquakes mean shorter intervals between them. Understanding this calculation is key to predicting future earthquake activities based on past patterns.