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Cumulative Distribution Function (CDF)

Definition / Introduction

  • The Cumulative Distribution Function (CDF) provides a unified way to describe the probability distribution of any random variable, whether it's Discrete or Continuous.
  • The CDF, denoted \(F(x)\) or \(F_X(x)\), gives the probability that the random variable \(X\) takes on a value less than or equal to a specific value \(x\).
  • It accumulates probability as the value \(x\) increases.

Key Concepts

1. Notation and Definition

  • The CDF of a random variable \(X\) is defined as: $$ F(x) = P(X \le x) $$
  • For any given value \(x\), \(F(x)\) tells you the total probability accumulated up to that point.

2. Properties of a CDF

A function \(F(x)\) is a valid CDF if and only if it satisfies these properties: * Non-decreasing: If \(a < b\), then \(F(a) \le F(b)\). As \(x\) increases, the accumulated probability cannot decrease. * Right-continuous: The function is continuous from the right: \(\lim_{t \to x^+} F(t) = F(x)\). (This is more technical, but important for precise definitions, especially with discrete variables). * Limits: * The limit as \(x\) approaches negative infinity is 0: \(\lim_{x \to -\infty} F(x) = 0\) (There's no accumulated probability infinitely far to the left). * The limit as \(x\) approaches positive infinity is 1: \(\lim_{x \to +\infty} F(x) = 1\) (Eventually, all probability is accumulated).

3. CDF for Discrete Random Variables

  • Calculated by summing the PMF values \(p(t)\) for all possible outcomes \(t\) less than or equal to \(x\). $$ F(x) = P(X \le x) = \sum_{t \le x} p(t) $$
  • The graph of a discrete CDF is a step function. It jumps at each value \(x\) where \(p(x) > 0\), and the size of the jump is equal to \(p(x)\). It's constant between jumps.
  • Example (Two Coin Flips, X = # Heads): PMF was \(p(0)=1/4, p(1)=1/2, p(2)=1/4\). $$ F(x) = \begin{cases} 0 & \text{if } x < 0 \ ¼ & \text{if } 0 \le x < 1 \ ¾ & \text{if } 1 \le x < 2 \ 1 & \text{if } x \ge 2 \end{cases} $$

4. CDF for Continuous Random Variables

  • Calculated by integrating the PDF \(f(t)\) from negative infinity up to \(x\). $$ F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) \, dt $$
  • The graph of a continuous CDF is a continuous, non-decreasing function that smoothly rises from 0 to 1.
  • The value of the PDF \(f(x)\) is the slope (derivative) of the CDF \(F(x)\): \(f(x) = \frac{d}{dx} F(x)\).
  • Example (Uniform Distribution on [0, 2]): PDF was \(f(t) = 1/2\) for \(0 \le t \le 2\). $$ F(x) = \begin{cases} 0 & \text{if } x < 0 \ \int_0^x \frac{1}{2} dt = \frac{x}{2} & \text{if } 0 \le x \le 2 \ 1 & \text{if } x > 2 \end{cases} $$ So, \(F(x)\) starts at 0, rises linearly to 1 between x=0 and x=2, and stays at 1 thereafter.

5. Usage: Calculating Probabilities from CDF

  • The CDF is very useful for finding the probability that \(X\) falls within any interval \((a, b]\): $$ P(a < X \le b) = F(b) - F(a) $$
  • This works for both discrete and continuous variables.
  • Other probabilities can also be derived:
    • \(P(X > a) = 1 - P(X \le a) = 1 - F(a)\)
    • \(P(X < a)\) is \(F(a)\) for continuous RVs, but \(F(a) - P(X=a)\) for discrete RVs (or \(\lim_{t \to a^-} F(t)\)).
    • \(P(X \ge a) = 1 - P(X < a)\).
    • \(P(X=a) = F(a) - \lim_{t \to a^-} F(t)\) (This is zero for continuous RVs, and equal to the jump size \(p(a)\) for discrete RVs).

Connections to Other Topics

  • Directly related to PMF (by summation) and PDF (by integration/differentiation).
  • Provides a complete description of a random variable's distribution.
  • Used to find percentiles (quantiles) of a distribution. The median, for example, is the value \(m\) such that \(F(m) = 0.5\).
  • Used in generating random numbers from specific distributions (Inverse Transform Sampling: if \(U \sim \text{Uniform}(0, 1)\), then \(X = F^{-1}(U)\) follows the distribution with CDF \(F\)).

Summary

  • The CDF, \(F(x) = P(X \le x)\), gives the cumulative probability up to value \(x\).
  • Defined for both discrete and continuous random variables.
  • Properties: Non-decreasing, right-continuous, limits 0 and 1.
  • Graph: Step function for discrete RVs, continuous curve for continuous RVs.
  • Useful for calculating probabilities over intervals: \(P(a < X \le b) = F(b) - F(a)\).

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