Cumulative Distribution Function

Definition

The Cumulative Distribution Function (CDF) describes the probability that a real-valued random variable $X$ will take a value less than or equal to a given number $x$. For a continuous random variable, the CDF is the integral of its Probability Density Function (PDF) from negative infinity up to $x$.

$F_X(x) = P(X \leq x) = \int_{-\infty}^{x} f_X(t) dt$ How to read: F sub X of x equals the probability that X is less than or equal to x, which equals the integral from negative infinity to x of f sub X of t with respect to t. Meaning / when to use: Used to find the total accumulated probability up to a certain point $x$. It completely characterizes the distribution of a random variable.

Why It Matters

While a Probability Density Function tells you the relative likelihood of landing exactly on a specific point, it doesn’t directly answer questions like “What is the probability of the system failing before 10,000 hours?” The CDF provides the direct mathematical tool to answer cumulative questions, which are the most common in risk management, reliability engineering, and actuarial science. It translates abstract density curves into actionable probabilities.

Core Concepts

• Monotonicity: The CDF is a non-decreasing function; as $x$ increases, the cumulative probability can only stay the same or grow.
• Bounds: The CDF always starts at 0 (as $x \to -\infty$) and approaches 1 (as $x \to \infty$).
• Relationship to PDF: The Fundamental Theorem of Calculus links them: the derivative of the CDF is the PDF ($f_X(x) = \frac{d}{dx}F_X(x)$).
• Percentiles: The inverse of the CDF allows you to calculate percentiles (e.g., finding the score below which 90% of observations fall).

Connected Concepts

• Central Limit Theorem
• Monotonicity
• Gaussian Integral
• Integration Constant