Question

Read Traffic Density. Function random0 produces a number with a uniform probability distribution in the range \([0.0,1.0)\). This function is suitable for simulating random events if each outcome has an equal probability of occurring. However, in many events the probability of occurrence is not equal for every event, and a uniform probability distribution is not suitable for simulating such events. For example, when traffic engineers studied the number of cars passing a given location in a time interval of length \(t\), they discovered that the probability of \(k\) cars passing during the interval is given by the equation $$ P(k, t)=e^{-\lambda t} \frac{(\lambda t)^{4}}{k !} \text { for } t \geq 0, \lambda>0, \text { and } k=0,1,2, \ldots $$ This probability distribution is known as the Poisson distribution: it occurs in many applications in science and engineering. For example, the number of calls \(k\) to a telephone switchboard in time interval \(t\), the number of bacteria \(k\) in a specified volume \(t\) of liquid, and the number of failures \(k\) of a complicated system in time interval \(t\) all have Poisson distributions. Write a function to evaluate the Poisson distribution for any \(k, t\), and A. Test your function by calculating the probability of \(0,1,2, \ldots, 5\) cars passing a particular point on a highway in I minute, given that \(\lambda\) is \(1.6\) per minute for that highway. Plot the Poisson distribution for \(t=1\) and \(\lambda=1.6\).

Short answer

To compute the Poisson distribution for a given \(k\), \(t\), and \(\lambda\), we can define a function using the formula \(P(k, t) = e^{-\lambda t} \frac{(\lambda t)^{k}}{k!}\) in Python: ```python import math def poisson_distribution(k, t, lmbda): return math.exp(-lmbda * t) * (lmbda * t)**k / math.factorial(k) ``` We test the function by calculating the probabilities of observing \(0, 1, 2, ..., 5\) cars within \(1\) minute for a highway with \(\lambda=1.6\), and plotting the results using the Matplotlib library: ```python import matplotlib.pyplot as plt lmbda = 1.6 t = 1 probabilities = {} for k in range(6): probabilities[k] = poisson_distribution(k, t, lmbda) x = list(probabilities.keys()) y = list(probabilities.values()) plt.bar(x, y) plt.xlabel('Number of Cars') plt.ylabel('Probability') plt.title('Poisson Distribution (t=1 minute, λ=1.6 per minute)') plt.show() ``` This code snippet computes and plots the Poisson distribution for \(t=1\) and \(\lambda=1.6\).

Step-by-step solution

Define the Poisson distribution formula

The Poisson distribution formula is given as: $$ P(k, t) = e^{-\lambda t} \frac{(\lambda t)^{k}}{k!} $$ Step 2: Implement the function for the Poisson distribution

Implement the function for the Poisson distribution

The Poisson distribution function can be implemented in Python as follows: ```python import math def poisson_distribution(k, t, lmbda): return math.exp(-lmbda * t) * (lmbda * t)**k / math.factorial(k) ``` Step 3: Calculate probabilities for given input data

Calculate probabilities for given input data

Using the provided function, we can now compute the probability of \(0, 1, 2, \ldots, 5\) cars passing a particular point on a highway in 1 minute, given that \(\lambda = 1.6\) per minute. We will store the results in a dictionary. ```python lmbda = 1.6 t = 1 probabilities = {} for k in range(6): probabilities[k] = poisson_distribution(k, t, lmbda) ``` Step 4: Plot the Poisson distribution for \(t=1\) and \(\lambda=1.6\)

Plot the Poisson distribution

To plot the Poisson distribution, we can use Python's Matplotlib library. First, you need to install the library with: ```sh pip install matplotlib ``` Then plot the Poisson distribution with: ```python import matplotlib.pyplot as plt x = list(probabilities.keys()) y = list(probabilities.values()) plt.bar(x, y) plt.xlabel('Number of Cars') plt.ylabel('Probability') plt.title('Poisson Distribution (t=1 minute, λ=1.6 per minute)') plt.show() ``` This plot will show the probability of each value of \(k\) (number of cars) within 1 minute for a highway with \(\lambda=1.6\).

Key concepts

MATLAB Programming

Understanding MATLAB programming is essential for engineers and scientists working on numerical analysis and data visualization. MATLAB, a high-level language and interactive environment, enables users to perform computationally intensive tasks faster than with traditional programming languages such as C, C++, and Fortran.

In relation to our exercise, MATLAB can be employed to simulate probability distributions like the Poisson distribution and visualize the results. The code structure in MATLAB is straightforward: you define functions, such as one for the Poisson distribution, and then call these functions with specific parameters to simulate different scenarios. For example, to simulate traffic density, one might do the following in MATLAB:
function P = poisson_distribution(k, t, lambda)
P = exp(-lambda * t) * (lambda * t)^k / factorial(k);
end
After defining the function, MATLAB can calculate probabilities for different numbers of cars passing and plot the results using its powerful plotting functions.

Traffic Density Simulation

Traffic density simulation is an application of probability theory in the field of traffic engineering, where the goal is to model and analyze the flow and density of traffic on roadways. By using MATLAB programming, simulations can be performed to predict the number of cars passing a point over time based on probability distributions.

The Poisson distribution, often used in these simulations, can help estimate the occurrence of rare, independent events within a fixed interval of time. For instance, traffic engineers might simulate the number of cars that pass a highway tollbooth in a minute to optimize toll collection or manage traffic flow. This type of simulation assists in designing road systems that can handle peak traffic loads efficiently and safely.

Probability Distributions

Probability distributions are foundational concepts in statistics and are used to describe the likelihood of various outcomes. The Poisson distribution is a particular type of probability distribution that is useful for modeling the number of times an event occurs within a specific interval of time or space.

It is particularly well-suited to engineering statistics for events that occur independently and at a constant average rate. Key parameters of the Poisson distribution include \( \lambda \), the average number of occurrences within the given interval. The formula provided in the original exercise, \( P(k, t) = e^{-\lambda t} \frac{(\lambda t)^{k}}{k!} \), captures these elements and allows for the calculation of the probability associated with various outcomes (\(k\) events). Understanding this distribution aids engineering students in analyzing patterns that range from traffic flows to signal processing.

Engineering Statistics

Engineering statistics involves applying statistical methods to engineering problems. It plays a critical role in quality control, reliability, and design of new products and systems. Statistics help engineers make informed decisions based on data and probabilities, often using distributions like the Poisson distribution for such decisions.

In the context of the original problem of analyzing traffic density, the Poisson distribution aids in predicting the variability and probability of events - such as the number of cars passing a point. The use of engineering statistics allows for optimizing systems to accommodate typical and atypical loads, helping to prevent system failures and design more robust infrastructures. Therefore, understanding and utilizing the Poisson distribution within the framework of engineering statistics empowers students and professionals to precisely quantify risks and design systems that are both effective and reliable.