Definition of the distribution function and its basic properties

Introduction

The distribution function, often denoted as $ F(x) $, is a fundamental concept in probability and statistics. It provides a comprehensive description of the probability distribution of a random variable $ X $. Specifically, $ F(x) $ is defined as the probability that $ X $ takes a value less than or equal to $ x $, mathematically expressed as $ F(x) = \mathbb{P}(X \leq x) $. This function serves as a bridge between the discrete and continuous probability models, enabling the analysis of random variables through their cumulative behavior. The distribution function is inherently tied to the probability density function (PDF), which characterizes the rate of change of the CDF. By understanding the properties of $ F(x) $, one can gain insight into the distributional characteristics of random variables, such as their concentration, spread, and tail behavior.

Formal Definition and Basic Properties

The distribution function $ F(x) $ is defined for all real numbers $ x $ and satisfies the following properties:

  1. Right-Continuity: $ F(x) $ is right-continuous, meaning $ \lim_{h \to 0^-} F(x + h) = F(x) $ for all $ x $.
  2. Non-Decreasing: $ F(x) $ is non-decreasing, ensuring that $ F(x_1) \leq F(x_2) $ whenever $ x_1 \leq x_2 $.
  3. Limits at Infinity: $ \lim_{x \to -\infty} F(x) = 0 $ and $ \lim_{x \to \infty} F(x) = 1 $.
  4. Measurability: $ F(x) $ is a measurable function, aligning with the principles of measure theory.

These properties ensure that $ F(x) $ captures the cumulative probability of the random variable's values, providing a complete picture of its distribution. The function $ F(x) $ is also closely related to the PDF $ f(x) $, which is the derivative of $ F(x) $ in the case of continuous variables. For discrete variables, $ F(x) $ is a step function that increases by $ \mathbb{P}(X = x) $ at each point $ x $, reflecting the probability mass at that value.

Key Properties of the Distribution Function

The distribution function $ F(x) $ possesses several critical properties that govern its behavior:

  • Continuity: While $ F(x) $ is right-continuous, it may have discontinuities at points where the random variable $ X $ takes on specific values. These discontinuities correspond to jumps in the probability mass, and the size of the jump is equal to $ \mathbb{P}(X = x) $.
  • Monotonicity: The non-decreasing nature of $ F(x) $ ensures that the probability of $ X $ being less than or equal to a larger value is always greater than or equal to that of a smaller value.
  • Integral Representation: The cumulative distribution function is defined through integration, with $ F(x) = \int_{-\infty}^{x} f(t) , dt $ for continuous random variables, where $ f(t) $ is the PDF. This relationship underscores the connection between the CDF and the PDF, which is central to understanding the distribution of $ X $.

The distribution function also satisfies the cumulative distribution theorem, which states that $ F(x) $ is uniquely determined by its values at the points of discontinuity and its behavior at infinity. This theorem ensures that $ F(x) $ is well-defined and consistent across different probability models.

Relationship with Probability Density Functions

The relationship between the distribution function and the PDF is pivotal in the analysis of random variables. For continuous random