What is Beta Distribution in Python

The Beta Distribution

The Beta distribution is a continuous probability distribution that models the probability of an event occurring between 0 and 1. It’s particularly useful in scenarios where you have some prior information about the likelihood of success or failure and want to represent that information probabilistically.

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Key Characteristics:

Defined on the interval [0, 1]
Controlled by two shape parameters, alpha (α) and beta (β), which affect the shape of the distribution
Commonly referred to as the “distribution of probabilities” because it can model probabilities themselves

Applications:

Bayesian Statistics: The Beta distribution is a popular choice for conjugate priors in Bayesian inference. A conjugate prior is a prior distribution that leads to a convenient posterior distribution when combined with a likelihood function.
Modeling Proportions: It can be used to model proportions, percentages, or rates that fall within the 0-1 range. For example, the probability of a customer making a purchase (conversion rate) or the proportion of time a machine spends in an operational state.
Smoothing Estimates: In situations with limited data, the Beta distribution can be used to “smooth” estimates by incorporating prior knowledge or beliefs.

Using the Beta Distribution in Python

Python offers several libraries for working with the Beta distribution:

SciPy (scipy.stats)

Provides functions for generating random samples, calculating the probability density function (PDF), cumulative distribution function (CDF), and other statistical properties.
Python
import scipy.stats as stats

# Example parameters
alpha = 2
beta = 3

# Generate random samples
samples = stats.beta.rvs(alpha, beta, size=1000)

# Calculate PDF at specific values
x = 0.4
pdf_value = stats.beta.pdf(x, alpha, beta)

# Calculate CDF at specific values
cdf_value = stats.beta.cdf(x, alpha, beta)
NumPy (numpy.random.beta)

Offers a function to generate random samples from the Beta distribution.
Python
import numpy as np

# Example parameters (same as SciPy)
alpha = 2
beta = 3

# Generate random samples
samples = np.random.beta(alpha, beta, size=1000)

Visualization

You can use Matplotlib to visualize the Beta distribution:

Python


import matplotlib.pyplot as plt
import scipy.stats as stats

# Example parameters
alpha = 2
beta = 3

# Create x-axis values
x = np.linspace(0, 1, 100)  # 100 points between 0 and 1

# Calculate PDF values
pdf = stats.beta.pdf(x, alpha, beta)

# Plot the PDF
plt.plot(x, pdf, label=f"Beta Distribution (alpha={alpha}, beta={beta})")
plt.xlabel("x")
plt.ylabel("Probability Density")
plt.title("Beta Distribution")
plt.legend()
plt.show()