Introduction:

In this blog post, we’ll take a detailed look at bivariate distributions using Python. Specifically, we’ll focus on a Monte Carlo random sampling approach to understand and generate random samples from this distribution.

Understanding the Process:

  1. Library Imports: We kickstart our exploration by importing essential libraries-NumPy and SciPy. These libraries empower us with the necessary tools for mathematical computations and statistical operations.

  2. Defining Domain Boundaries: To set the stage, we define the boundaries of our analysis-namely, the specific regions of interest within the bivariate normal distribution. Descriptive variables like x_left, x_right, y_bottom, and y_top establish these boundaries.

  3. The calculate_intensity Function with Slicing: At the core of our journey is the calculate_intensity function, responsible for modeling the bivariate normal distribution. It calculates the distribution’s intensity at different points within the domain, taking parameters like mean, covariance_matrix, and scaling_factor into account.

    What sets this function apart is the “slicing” mechanism. It selectively zeroes out intensity values in specific regions, adding complexity and realism to the distribution. This slicing creates sharp transitions in intensity, making it more interesting and applicable to real-world scenarios.

  4. Visualizing the Distribution: To gain insights into the distribution’s behavior, we turn to the plot_distribution function. It enables us to visualize the bivariate normal distribution vividly. By generating a grid of (x, y) points within the domain, we create a canvas that visually represents intensity variations. These intensity values come from the calculate_intensity function. Using Matplotlib, we craft an engaging 3D surface plot that brings the distribution to life.

  5. Calculating the Integral (\(\mu\)): Our journey takes a statistical turn as we calculate the integral of the intensity function over the defined domain. This integral result represents the statistical parameter (\(\mu\)), signifying the average number of events in a Poisson process-a crucial insight into the distribution’s real-world implications.

  6. Monte Carlo Random Sampling:
    • We step into the realm of Monte Carlo random sampling. Here, we simulate a random sample size (sample_size) from a Poisson distribution. This simulated sample mimics the number of events in a Poisson process.
    • Our quest for parameter \(\alpha\) (alpha) unfolds through optimization techniques. By minimizing the negative intensity function, we effectively maximize the distribution’s highest intensity value.

  7. Accept-Reject Sampling (The “generate_sample” Function): We delve into the intriguing world of accept-reject sampling to obtain a random sample from the bivariate normal distribution. This technique involves accepting or rejecting points based on their intensity values, ensuring our sample accurately reflects the distribution’s characteristics. It includes scaling the sample size by an efficiency factor (eff), generating random values for thresholds, x_candidates, and y_candidates, and conducting the acceptance or rejection process.

  8. Examining the Results: As our journey culminates, we present the results of our exploration. This includes displaying the number of points in the generated sample and visualizing these points on a scatter plot. These visualizations provide a tangible representation of the random sample drawn from the bivariate normal distribution, offering insights into the practical implications of our analysis.

Conclusion:

This Python script serves as your guide through the intricate world of bivariate normal distributions, employing Monte Carlo random sampling. The “slicing” technique within the calculate_intensity function adds depth and realism to the distribution, making it a valuable tool for understanding statistical concepts. You can use this script as a foundation to enhance your understanding of probability distributions and statistical analysis using Python.