Computing Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a fundamental concept in probability statistics, representing the probability that a random variable takes a value less than or equal to a specific threshold. MATLAB provides several approaches to compute CDF, with two primary implementation strategies detailed below.

### Method 1: Empirical Distribution-Based Approach The first method calculates empirical CDF directly from sample data using MATLAB's built-in `ecdf` function. This function automatically computes the empirical cumulative distribution and returns two key outputs: sorted sample points and their corresponding cumulative probabilities. The implementation requires simply passing the data vector to `ecdf`, making it ideal for exploratory analysis when the theoretical distribution is unknown. The empirical approach provides a data-driven visualization of cumulative distribution trends without requiring distribution assumptions.

### Method 2: Theoretical Distribution-Based Approach The second method applies when the random variable follows a known theoretical distribution (e.g., normal, Poisson). MATLAB's Statistics and Machine Learning Toolbox offers specialized functions like `normcdf` for normal distributions and `poisscdf` for Poisson distributions. These functions require specifying distribution parameters (mean, variance for normal; lambda for Poisson) and return precise cumulative probabilities. The implementation involves calling the appropriate distribution-specific function with parameters and evaluation points, providing smooth, efficient calculations for known distributions.

Each method has distinct advantages: empirical CDF requires no distribution assumptions and suits exploratory analysis, while theoretical CDF offers computational efficiency and smooth results for modeling scenarios with known distributions. The choice depends on data characteristics and application requirements, with empirical methods favoring data exploration and theoretical methods preferred for parametric modeling.