Problem 56
Question
Assume that \(N(t)\) is a Poisson process with rate \(\lambda\) and \(T_{1}\) is the
time of the first arrival. Show that, for \(s
Step-by-Step Solution
Verified Answer
The probability \(P(T
1Step 1: Define the Conditional Probability
We need to find the conditional probability \(P(T_1 < s \mid N(t) = 1)\). This is the probability that the first arrival time \(T_1\) is less than \(s\), given that exactly one arrival occurs by time \(t\).
2Step 2: Use the Definition of Conditional Probability
According to the definition of conditional probability, \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\). Here, \(A\) is the event that \(T_1 < s\) and \(B\) is the event that \(N(t) = 1\).
3Step 3: Calculate \(P(T_1 < s \cap N(t) = 1)\)
For the event \(T_1 < s\) and exactly one event by time \(t\), \(N(s) = 1\) and no events occur between \(s\) and \(t\). Thus, \(P(T_1 < s \cap N(t) = 1) = P(N(s) = 1) \cdot P(N(t) - N(s) = 0)\).
4Step 4: Apply Poisson Process Probabilities
For a Poisson process, \(P(N(s) = 1) = \lambda s e^{-\lambda s}\) and \(P(N(t) - N(s) = 0) = e^{-\lambda (t-s)}\). Hence, \(P(T_1 < s \cap N(t) = 1) = \lambda s e^{-\lambda s} \cdot e^{-\lambda (t-s)} = \lambda s e^{-\lambda t}\).
5Step 5: Calculate \(P(N(t) = 1)\)
\(P(N(t) = 1) = \lambda t e^{-\lambda t}\), since exactly one event occurs by time \(t\) in a Poisson process.
6Step 6: Combine the Probabilities
Plug these into the conditional probability formula: \(P(T_1 < s \mid N(t) = 1) = \frac{\lambda s e^{-\lambda t}}{\lambda t e^{-\lambda t}} = \frac{s}{t}\).
7Step 7: Interpret the Result
This shows that given exactly one arrival by time \(t\), the arrival time is uniformly distributed over \([0, t)\). Hence, \(P(T_1 < s \mid N(t) = 1) = \frac{s}{t}\) confirms this uniform distribution.
Key Concepts
Conditional Probability in Poisson ProcessesUniform Distribution in Arrival TimeExploring Arrival Time in Poisson Processes
Conditional Probability in Poisson Processes
Conditional probability helps us understand the likelihood of an event occurring, given that another event has already happened. In the context of Poisson processes, which deal with events happening independently over time, this concept is crucial. When we say we want to find \(P(T< s \mid N(t)=1)\), we are determining the probability that the first arrival occurs before time \(s\), given that exactly one arrival occurs by time \(t\). Here:
By focusing on how multiple conditions interact, conditional probability offers a powerful tool to handle complex problems, especially in statistical contexts like Poisson processes.
- \(T
- \(N(t)=1\) means exactly one arrival happens by time \(t\).
By focusing on how multiple conditions interact, conditional probability offers a powerful tool to handle complex problems, especially in statistical contexts like Poisson processes.
Uniform Distribution in Arrival Time
In some probability problems, after setting a condition, the outcomes distribute evenly within a given range. This is known as uniform distribution. In our Poisson process example, once we know that one event has occurred by time \(t\), the exact timing of that event between \([0,t)\) is uniformly distributed.
This means that every moment within the interval is equally likely to be the arrival time. The phrase \(P(T_1 < s \mid N(t) = 1)=\frac{s}{t}\) illustrates precisely this concept. It tells us that if one arrival happens by \(t\), the probability that it happened by \(s\) is proportional to the time divided by the total interval length, \(\frac{s}{t}\).
This means that every moment within the interval is equally likely to be the arrival time. The phrase \(P(T_1 < s \mid N(t) = 1)=\frac{s}{t}\) illustrates precisely this concept. It tells us that if one arrival happens by \(t\), the probability that it happened by \(s\) is proportional to the time divided by the total interval length, \(\frac{s}{t}\).
- This simplification comes from the nature of the Poisson process, where arrivals are independent and randomly distributed over time intervals.
- It highlights why using distributions, like uniform, simplifies math problems by presenting each outcome with equal opportunity.
Exploring Arrival Time in Poisson Processes
Arrival time, specifically the first arrival time \(T_1\), in a Poisson process is a fundamental aspect. Poisson processes model random events occurring over a continuous timeline. The focus on \(T_1\) typically arises because knowing the time of the first event helps in modeling or predicting future events.
In examining \(T_1 < s\), we study how early the first event could occur in a given time window, \([0, s)\). In Poisson processes, another key feature is the exponential distribution that typically follows: the time until the first event occurs is exponentially distributed when events happen independently and sporadically.
In examining \(T_1 < s\), we study how early the first event could occur in a given time window, \([0, s)\). In Poisson processes, another key feature is the exponential distribution that typically follows: the time until the first event occurs is exponentially distributed when events happen independently and sporadically.
- \(T_1\), as a first occurrence, marks the beginning of the event timeline in a process.
- The analysis often simplifies subsequent predictions and computations in statistical models.
- This all demonstrates how focusing on arrival time, such as \(T_1\), aids in creating structures for event prediction in fields like physics or telecommunications.
Other exercises in this chapter
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