Suppose that every three months, on average, an earthquake occurs in a certain region. Assuming this is a Poisson process, what is the probability that the next earthquake occurs after three but before seven months?
Suppose that every three months, on average, an earthquake occurs in a certain region. Assuming this is a Poisson process, what is the probability that the next earthquake occurs after three but before seven months?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![**Poisson Process Probability Analysis**
**Question:**
Suppose that every three months, on average, an earthquake occurs in a certain region. Assuming this is a Poisson process, what is the probability that the next earthquake occurs after three but before seven months?
**Explanation:**
To solve this problem, we need to determine the probability in a Poisson process context where an event (earthquake) occurs at a certain rate. Here, we are given that an earthquake occurs once every three months on average.
To find the probability that the next earthquake occurs between three and seven months, we employ the exponential distribution, which is used to model the time between events in a Poisson process. The rate \( \lambda \) of occurrence (in events per month) is \( 1/3 \).
The probability that the time \( T \) till the next earthquake occurs after a given time is \( P(T > t) = e^{-\lambda t} \). To find \( P(3 < T < 7) \), we calculate:
\[ P(3 < T < 7) = P(T < 7) - P(T < 3) \]
The cumulative distribution function (CDF) for the exponential distribution gives:
\[ P(T < t) = 1 - e^{-\lambda t} \]
Substitute the values for \( t=3 \) and \( t=7 \) to find the required probability:
\[ P(T < 7) = 1 - e^{-\frac{7}{3}} \]
\[ P(T < 3) = 1 - e^{-1} \]
Thus, the probability \( P(3 < T < 7) \) is the difference:
\[ P(3 < T < 7) = (1 - e^{-\frac{7}{3}}) - (1 - e^{-1}) \]
These computations provide the probability for the earthquake occurring in the specified time frame.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F522630ab-9263-4897-b338-a0fdea344135%2F9773da61-bd30-40b1-a879-30816b063bf7%2F34dhuqg_processed.png&w=3840&q=75)
Transcribed Image Text:**Poisson Process Probability Analysis**
**Question:**
Suppose that every three months, on average, an earthquake occurs in a certain region. Assuming this is a Poisson process, what is the probability that the next earthquake occurs after three but before seven months?
**Explanation:**
To solve this problem, we need to determine the probability in a Poisson process context where an event (earthquake) occurs at a certain rate. Here, we are given that an earthquake occurs once every three months on average.
To find the probability that the next earthquake occurs between three and seven months, we employ the exponential distribution, which is used to model the time between events in a Poisson process. The rate \( \lambda \) of occurrence (in events per month) is \( 1/3 \).
The probability that the time \( T \) till the next earthquake occurs after a given time is \( P(T > t) = e^{-\lambda t} \). To find \( P(3 < T < 7) \), we calculate:
\[ P(3 < T < 7) = P(T < 7) - P(T < 3) \]
The cumulative distribution function (CDF) for the exponential distribution gives:
\[ P(T < t) = 1 - e^{-\lambda t} \]
Substitute the values for \( t=3 \) and \( t=7 \) to find the required probability:
\[ P(T < 7) = 1 - e^{-\frac{7}{3}} \]
\[ P(T < 3) = 1 - e^{-1} \]
Thus, the probability \( P(3 < T < 7) \) is the difference:
\[ P(3 < T < 7) = (1 - e^{-\frac{7}{3}}) - (1 - e^{-1}) \]
These computations provide the probability for the earthquake occurring in the specified time frame.
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