Problem 55 - Calculus for Biology and Medicine

Question

You arrive at a bus stop at a random time. Assuming that busses arrive according to a Poisson process with rate \(4 / \mathrm{hr}\), what is the expected time to the next arrival?

Step-by-Step Solution

Answer

The expected time for the next bus is 15 minutes.

Step 1: Understand the Poisson Process

A Poisson process is one where events happen continuously and independently at a constant average rate. For this problem, buses arrive at a rate of \(\lambda = 4\) buses per hour.

Step 2: Define Expected Waiting Time

The problem asks for the expected time you need to wait for the next bus. In a Poisson process, the time between events is exponentially distributed with parameter \(\lambda\). The expected waiting time, \(E[T]\), where \(T\) is the time to the next event, is given by \(1/\lambda\).

Step 3: Calculate the Expected Waiting Time

Given that \(\lambda = 4\) buses per hour, the expected time until the next bus arrives is calculated as: \[ E[T] = \frac{1}{\lambda} = \frac{1}{4} \text{ hours} \] Convert this into minutes (since 1 hour = 60 minutes): \[ E[T] = \frac{60}{4} = 15 \text{ minutes} \]

Key Concepts

Expected Waiting Time

When we talk about expected waiting time, we are referring to the average time you anticipate waiting for a certain event, like a bus arrival. In a Poisson process, where events occur randomly and independently at an average rate, this expected waiting time can be calculated using a neat mathematical formula.
To find this expected waiting time, we look at the average rate at which events occur, known as the rate parameter, denoted by \( \lambda \). For a Poisson process, the time between consecutive arrivals follows an exponential distribution. Thus, the expected waiting time, \( E[T] \), which tells us how long we will typically wait for the next event, calculates as \( \frac{1}{\lambda} \).
In practical terms, knowing the expected waiting time helps manage time better and is handy in planning. For instance, if you know a bus arrives on average every 15 minutes, there's less stress over how long you might wait at the stop. Remember, the _"expected"_ time doesn't mean you will always wait exactly that long, but it's an average you can expect over many instances.

Exponential Distribution

Exponential distribution might sound complex, but it's just a fancy way of saying the time between events (like your bus arriving) is spread out in a certain way. This is common in situations where events happen independently and continuously.
The exponential distribution is closely related to the Poisson process since it describes the interval of time expected between these randomly occurring events. So, when the rate parameter \( \lambda \) is 4, this means on average, 4 events (in this case, buses) occur per hour. Each bus arrival varies in time, but the average time to wait stays consistent. It's like a balancing act!
What makes the exponential distribution cool is its memoryless property. This means the probability of an event occurring in the future is independent of any past events. It's quite fascinating because it resets after every event. So, whether you've waited five minutes or just arrived, the chance of the bus showing up remains the same.

Rate Parameter

The rate parameter, symbolized as \( \lambda \), plays a crucial role in understanding both the Poisson process and the exponential distribution. It tells us how frequently events, like bus arrivals, happen in a given timeframe, which in our example, is per hour.
Think of \( \lambda \) as a measure of regularity: a higher value indicates more frequent arrivals, while a lower one suggests rarer events. In our bus example, \( \lambda = 4 \) implies you should expect around four buses every hour.
Knowing the value of \( \lambda \) allows us to compute various probabilities and averages, like our expected waiting time, \( E[T] \), using the formula \( \frac{1}{\lambda} \). This parameter is essential for modeling real-world scenarios where arrangements or predictions depend on average occurrences, making it a central piece in stochastic processes and studies.