## 
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Probability theory introduces the concept of a **probability density function (PDF)**, which expresses the likelihood of a continuous random variable taking on a particular value. We can leverage powerful libraries like NumPy, SciPy, and Matplotlib to plot the PDF of a continuous random variable in Python. The mathematical model for the PDF is as follows:
$$
F(X) = P(a \le x \le b) = \int_{a}^{b} f(x) dx \ge 0
$$
 
This equation, which expresses the likelihood that $X$ lies inside the range $[a, b]$, is created by integrating the PDF $f(x)$ over the interval.

The PDF $f(x)$ has two essential criteria:

1. **Non-negativity:** The PDF shouldn't be negative to all values of $x$ inside the range of $X$.
The probability values supplied by the PDF are always non-negative.

2. **Normalization:** Over the whole range of $X$, the PDF integrates to 1. This characteristic guarantees that the overall probability for the entire range of $X$ is 1. It denotes that a value within the scope of the random variable $X$ will permanently be assigned to it.

## Example
For instance, look at this code snippet that utilizes NumPy and Matplotlib to visualize the PDF of a normal distribution.

import numpy as np
import matplotlib.pyplot as plt

# Define the parameters of the normal distribution
mu = 0  # mean
sigma = 1  # standard deviation

# Generate random numbers from the normal distribution
x = np.random.normal(mu, sigma, 10000)

# Plot the PDF of the normal distribution
plt.hist(x, bins=50, density=True, alpha=0.6, color='g')
plt.xlabel('Value')
plt.ylabel('Probability density')
plt.title('Normal distribution PDF')
plt.show()

To generate a histogram plot of a normal distribution's probability density function (PDF), we use the following steps:
- **Lines 5–6:** We define the mean and standard deviation parameters using the `mu` and `sigma` variables. 

- **Line 9:** We use the `np.random.normal()` function to generate 10,000 random numbers from the distribution.

- **Line 12:** We use the `plt.hist()` function from `matplotlib` with the following parameters to plot the PDF:
  - `density=True`: This normalizes the histogram area under the curve to represent a probability density.
  - `alpha`: This is used to adjust the transparency level of the histogram bars.
- **Lines 13–15:** We add the labels and title to the graph.

The histogram will show a bell curve centered around the mean (`mu`) with a spread determined by the standard deviation (`sigma`). This code generates a professional quality plot portraying the PDF of a normal distribution.

How to implement probability density function using Python

Use NumPy and Matplotlib to plot a PDF of a normal distribution by generating random numbers and visualizing with a normalized histogram.