Generating Random Variables from a Uniform Distribution

Generating Random Variables from a Uniform Distribution#

A common and general technique for generating random variables with a specified distribution starts with a uniformly distributed random variable $U$ over the interval $[0, 1]$ .

To transform $U$ into a random variable $X$ with the desired distribution, use the following formula:

X = F_{X}^{- 1} (U)

where $F_{X}$ is the cumulative distribution function (CDF) of $X$ , and $F_{X}^{- 1}$ is the inverse of the CDF.

Example: Generating an Exponential Random Variable from a Uniform Distribution#

To generate an exponential random variable from a uniform distribution, we follow these steps:

Start with a Uniform Random Variable
Let $U$ be a random variable uniformly distributed over the interval $[0, 1]$ , denoted as $U \sim U (0, 1)$ . This means that $U$ can take any value in $[0, 1]$ with equal probability.

Define the PDF of the Target Distribution
Assume we want to generate a random variable $X$ that follows an exponential distribution. The probability density function (PDF) of an exponentially distributed random variable with rate parameter $λ = 1$ is:

f_{X} (x) = e^{- x}, x \geq 0

This PDF describes the likelihood of different values of $X$ occurring. The parameter $λ$ determines the rate at which the distribution decays. For simplicity, we assume $λ = 1$ .

Determine the CDF of the Target Distribution
The cumulative distribution function (CDF) $F_{X} (x)$ gives the probability that the random variable $X$ is less than or equal to a specific value $x$ . For an exponential distribution, the CDF is:

F_{X} (x) = 1 - e^{- x}, x \geq 0

This function represents the cumulative probability up to a point $x$ , obtained by integrating the PDF.

Use the Inverse CDF Transformation
To generate a value of $X$ from a given uniform random variable $U$ , we apply the inverse CDF transformation. This method transforms the uniform random variable into a random variable with the desired distribution.

When using the inverse CDF transformation, we equate the cumulative distribution function (CDF) of the desired random variable $X$ to a realization of the uniform random variable $U$ , i.e., $U \sim U (0, 1)$ .

In mathematical terms:

F_{X} (x) = u,

where $u$ is a specific value (a realization) of $U$ drawn from the uniform distribution $U (0, 1)$ . This equality ensures that $X$ will have the desired distribution since the uniform random variable $U$ is transformed via the inverse of $F_{X} (x)$ .

For this example, we set the CDF of $X$ equal to $u$ and solve for $x$ :

u = F_{X} (x) = 1 - e^{- x}

Solving this equation for $x$ gives the transformation that maps the uniform random variable $U$ to the exponential random variable $X$ . This process guarantees that $X$ will follow the exponential distribution.

Rearranging the equation to solve for $x$ :

e^{- x} = 1 - u

- x = \ln (1 - u)

x = - \ln (1 - u)

Thus, the transformation $x = - \ln (1 - u)$ converts the uniformly distributed random variable $U$ into an exponentially distributed random variable $X$ .

Generate the Exponential Random Variable
To generate a sample $x$ :

First, we generate a sample $U$ from the uniform distribution $U (0, 1)$ .
We apply the transformation $x = - \ln (1 - u)$ to obtain the corresponding value from the exponential distribution.

This process can be repeated to generate multiple samples from the exponential distribution, with each sample derived from a new independent uniform random variable $U$ .

Simulation#

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon

# Set the seed for reproducibility
np.random.seed(0)

# Number of samples to generate
n_samples = 10000

# Method 1: Generate from uniform distribution and transform using inverse CDF
u = np.random.uniform(0, 1, n_samples)
x_empirical = -np.log(1 - u)

# Method 2: Using built-in exponential distribution generator
x_builtin = np.random.exponential(scale=1.0, size=n_samples)

# Plot the histograms
plt.figure(figsize=(12, 6))

# Histogram of empirical x
plt.hist(x_empirical, bins=50, density=True, alpha=0.3, label='Empirical $x$ from $u$')

# Histogram of built-in x
plt.hist(x_builtin, bins=100, density=True, alpha=0.3, label='Built-in $x$ from numpy')


# Generate the x values
x_vals = np.linspace(0, 10, 1000)

# Theoretical PDF using scipy
pdf_scipy = expon.pdf(x_vals)

# Theoretical PDF using the expression p_x(x) = e^{-x}
pdf_expr = np.exp(-x_vals)

# Plot the scipy.stats.expon PDF
plt.plot(x_vals, pdf_scipy, 'b-', lw=2, label='Theoretical PDF (scipy.stats.expon)')

# Plot the PDF using the expression p_x(x) = e^{-x}
plt.plot(x_vals, pdf_expr, 'r--', lw=2, label='$f_{X}(x) = e^{-x}$')

# Labels and title
plt.xlabel('x')
plt.ylabel('Density')
plt.title('Comparison of Exponential Distribution Generation Methods')
plt.legend()
plt.grid(True)
plt.show()

_images/a2c3ad19690312eeda2088bc5453412e4e978fccca3c4c0424f3ab4e377d700a.png

Schonhoff’s Method#

import numpy as np

def exprv(num):
    """
    Generate exponentially distributed random variables.

    Parameters:
    num (int): Number of random variables to generate.

    Returns:
    numpy.ndarray: Array of exponentially distributed random variables.
    """
    # Set the random seed for reproducibility
    np.random.seed(0)
    
    # Generate uniformly distributed random variables in the range (0, 1)
    u = np.random.rand(num)
    
    # Transform to exponentially distributed random variables using -log(u)
    ex = -np.log(u)
    
    return ex

# Example usage
num = 5  # Number of random variables to generate
exponential_rvs = exprv(num)
print("Exponentially distributed random variables:", exponential_rvs)

Exponentially distributed random variables: [0.5999966  0.33520792 0.50623057 0.60718385 0.85883631]

import numpy as np
import matplotlib.pyplot as plt

def ete(num):
    """
    Theoretical and Empirical PDFs of Exponentially distributed random numbers.

    Parameters:
    num (int): Number of random variables (RVs) to generate.

    Outputs:
    Displays the theoretical PDF and the empirical PDF.
    """
    # Generate uniformly distributed random variables
    u = np.random.rand(num)
    
    # Convert to exponentially distributed random variables
    z = -np.log(u)
    
    # Create a range for x values
    x = np.arange(0, 10.1, 0.1)
    
    # Compute the empirical PDF using histogram
    et, bin_edges = np.histogram(z, bins=x, density=False)
    
    # Normalize the empirical PDF
    n10 = num / 10
    nznorm = et / n10
    
    # Plot the empirical PDF
    plt.plot(bin_edges[:-1], nznorm, 'b', label='Empirical PDF')
    
    # Compute and plot the theoretical PDF
    y = np.exp(-x)
    plt.plot(x, y, 'r', label='Theoretical PDF')
    
    # Add labels, grid, and legend
    plt.xlabel('x')
    plt.ylabel('PDF')
    plt.grid(True)
    plt.legend()
    
    # Display the plot
    plt.show()

# Example usage
ete(1000)  # Generate and plot the PDFs using 1000 random variables

_images/79210711a31d8426069104db341d99a5d0729e1d69a6664816e034bfc4557515.png

References#

T. Schonhoff and A. Giordano, Detection and Estimation Theory and its Applications. Prentice Hall, 2006.