python numpy gamma distribution

Python Numpy Gamma Distribution

Introduction

In this article, I will guide you on how to implement the gamma distribution in Python using the Numpy library. The gamma distribution is a continuous probability distribution that is often used to model the time required to wait for a given number of events in a Poisson process.

Steps to Implement Gamma Distribution

Here is the step-by-step process to implement the gamma distribution in Python using the Numpy library:

Step	Description
Step 1	Import the required libraries
Step 2	Define the shape and scale parameters for the gamma distribution
Step 3	Generate random samples from the gamma distribution
Step 4	Plot the histogram of the generated samples
Step 5	Fit a gamma distribution to the generated samples
Step 6	Plot the probability density function (PDF) of the fitted gamma distribution
Step 7	Calculate the cumulative distribution function (CDF) of the fitted gamma distribution

Now, let's go through each step in detail.

Step 1: Import the required libraries

First, we need to import the necessary libraries for our implementation. We will be using the Numpy library for generating random numbers and performing mathematical operations, and the Matplotlib library for plotting the histogram and probability density function.

import numpy as np
import matplotlib.pyplot as plt

Step 2: Define the shape and scale parameters for the gamma distribution

Next, we need to define the shape and scale parameters for the gamma distribution. The shape parameter, denoted by 'k', determines the shape of the distribution, while the scale parameter, denoted by 'theta', determines the spread of the distribution.

shape = 2.5
scale = 1.0

Step 3: Generate random samples from the gamma distribution

Using the Numpy library, we can generate random samples from the gamma distribution by using the numpy.random.gamma function. We need to pass the shape and scale parameters, as well as the size of the sample we want to generate.

samples = np.random.gamma(shape, scale, size=1000)

Step 4: Plot the histogram of the generated samples

To visualize the distribution of the generated samples, we can plot a histogram using the Matplotlib library. The histogram represents the frequency of occurrence for different values in the sample.

plt.hist(samples, bins=30, density=True, alpha=0.75)
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.title('Histogram of Gamma Distribution')
plt.show()

Step 5: Fit a gamma distribution to the generated samples

To estimate the parameters of the gamma distribution that best fit our generated samples, we can use the numpy.random.gamma.fit function. This function returns the shape and scale parameters of the fitted distribution.

fit_shape, fit_scale = np.random.gamma.fit(samples)

Step 6: Plot the probability density function (PDF) of the fitted gamma distribution

To visualize the PDF of the fitted gamma distribution, we can plot a line graph using the numpy.random.gamma.pdf function. This function takes the x-values, shape parameter, and scale parameter as inputs and returns the corresponding y-values.

x = np.linspace(0, 10, 100)
y = np.random.gamma.pdf(x, fit_shape, fit_scale)

plt.plot(x, y)
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.title('Probability Density Function of Fitted Gamma Distribution')
plt.show()

Step 7: Calculate the cumulative distribution function (CDF) of the fitted gamma distribution

The cumulative distribution function (CDF) represents the probability that a random variable takes on a value less than or equal to a given value. We can calculate the CDF of the fitted gamma distribution using the numpy.random.gamma.cdf function.

cdf = np.random.gamma.cdf(x, fit_shape, fit_scale)

plt.plot(x, cdf)
plt.xlabel('Value')
plt.ylabel('Cumulative Probability')
plt.title('Cumulative Distribution Function of Fitted Gamma Distribution')
plt.show()

Conclusion

In this article, we have discussed how to implement the gamma distribution in Python using the Numpy library. We have covered the step-by-step process along with the corresponding code and explanations. By following these steps, you can generate random samples from the gamma distribution, plot the histogram, fit a gamma distribution, and visualize the PDF and CDF. The gamma distribution is a powerful tool for modeling various real-world phenomena, and understanding its implementation in Python can be beneficial for data analysis and statistical modeling.