Question

Computer plot on the same axes the normal density functions with \(\mu=0\) and \(\sigma=1\), 2, and 5. Label each curve with its \(\sigma\).

Short answer

Plot the normal density functions with σ=1, 2, and 5 using Matplotlib. Label each curve accordingly.

Step-by-step solution

Understand the Normal Density Function

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Set Up the Matplotlib Environment

To plot the normal density functions, set up the plotting environment using Matplotlib. First, import the necessary libraries.```pythonimport matplotlib.pyplot as pltimport numpy as npfrom scipy.stats import norm```

Define the x-axis Values

Define the range of x-values over which the normal density functions will be evaluated. A typical range for normal distributions is -10 to 10.```pythonx = np.linspace(-10, 10, 1000) # 1000 points from -10 to 10```

Compute y-axis Values for Each σ

For each σ (1, 2, 5), compute the corresponding y-values of the normal density function using the norm.pdf method from scipy.stats.```pythony1 = norm.pdf(x, 0, 1) # σ=1y2 = norm.pdf(x, 0, 2) # σ=2y5 = norm.pdf(x, 0, 5) # σ=5```

Plot the Curves

Use Matplotlib to plot the normal density functions on the same axes. Label each curve with its corresponding σ.```pythonplt.plot(x, y1, label='σ=1')plt.plot(x, y2, label='σ=2')plt.plot(x, y5, label='σ=5')plt.xlabel('x')plt.ylabel('Density')plt.legend()plt.title('Normal Density Functions')plt.show()```

Key concepts

Matplotlib

Matplotlib is a powerful plotting library in Python, commonly used for creating static, interactive, and animated visualizations. It's highly versatile and allows users to generate plots, histograms, bar charts, and more.
To use Matplotlib, you typically start by importing its `pyplot` module as follows:
```python
import matplotlib.pyplot as plt
```
This import statement lets you create a variety of plots easily. For example, you can create line plots, scatter plots, and histograms. Each plot can be customized with labels, titles, and legends to make it easy to understand the data.
For the given exercise, Matplotlib helps in plotting the normal density functions for different standard deviations (\(\textbf{σ} \)). By plotting these functions on the same axes, we can visually compare how the spread of the data changes with different values of \(\textbf{σ}\).

Python Programming

Python is a popular programming language known for its readability and broad application range in data analysis, web development, automation, and scientific computing. It provides numerous libraries that simplify complex tasks.
In the context of this exercise, Python serves as the primary tool for setting up the plotting environment, defining functions, and generating the required plots.
The code starts by importing essential libraries: `matplotlib.pyplot` for plotting, `numpy` for numerical operations, and `scipy.stats` for statistical functions. Next, it defines the x-axis values (a range of values from -10 to 10) and computes the corresponding y-values using the normal density function (provided by `norm.pdf`).
Here’s a quick example of how to set up an array of x-values using `numpy`:
```python
x = np.linspace(-10, 10, 1000)
```
This line generates 1,000 evenly spaced points between -10 and 10. These points are then used to evaluate the normal density function.

Probability Density Function

A Probability Density Function (PDF) describes the relative likelihood for a continuous random variable to take on a given value. For the normal distribution, the PDF is given by the formula:
\[\text{PDF}(x) = \frac{1}{\text{σ}\text{√}(2π)} \text{e}^{-\frac{(x - μ)^2}{2σ^2}}\]
Here, \(μ\) is the mean and \(σ\) is the standard deviation. This formula creates the familiar bell curve associated with normal distributions.
In our exercise, we visualize PDFs for different standard deviations (1, 2, and 5) while keeping the mean (\