1. Definition

For a random variable $X$, the cumulative distribution function (CDF) is:

$F(x) = P(X \leq x)$

  • It gives the probability that the variable takes a value less than or equal to $x$.
  • Defined for all real numbers $x$.
  • Range: $0 \leq F(x) \leq 1$.

2. Properties

  1. Non-decreasing: $F(x)$ never decreases as $x$ increases.
  2. Limits:
    • $\lim_{x \to -\infty} F(x) = 0$
    • $\lim_{x \to +\infty} F(x) = 1$
  3. Right-continuous: No sudden jumps except for discrete variables.

3. Relationship with PDF/PMF

  • If $X$ is continuous with density $f(x)$: $F(x) = \int_{-\infty}^x f(t)\,dt$ (Derivative: $f(x) = F'(x)$).
  • If $X$ is discrete with probability mass function $p(x)$): $F(x) = \sum_{t \leq x} p(t)$

4. Examples

Discrete Example

Let $X$ = dice roll {1,2,3,4,5,6}, each with $p=1/6$.

  • $F(3) = P(X \leq 3) = 1/6 + 1/6 + 1/6 = 0.5$.
  • $F(6) = 1.0$.

Continuous Example

If $X \sim N(0,1)$ (standard normal):

  • $F(0) = 0.5$.
  • $F(1.645) \approx 0.95$.
  • Meaning: 95% of values lie below 1.645.

5. Applications

  • Probabilities: To find chance of being within an interval: $P(a < X \leq b) = F(b) – F(a)$
  • Percentiles / Quantiles:
    • 25th percentile = smallest $x$ such that $F(x) = 0.25$.
    • Median = 50th percentile = $F(x)=0.5$.
  • Simulation: Generate samples by inverting the CDF.
  • Forecasting: Prediction intervals are based on quantiles of the forecast CDF.

6. Visualization

  • Discrete CDF → step function.
  • Continuous CDF → smooth S-shaped (sigmoid-like) curve.

Example: Standard normal CDF is symmetric and smooth, crossing 0.5 at $x=0$.

Summary:
The Cumulative Distribution Function (CDF) $F(x) = P(X \leq x)$ gives the probability that a random variable is less than or equal to a value. It’s non-decreasing, ranges from 0 to 1, and is central for computing probabilities, quantiles, and prediction intervals.