CDF

CDF. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable $X$, evaluated at a point $x$, is the probability that $X$ will take a value less than or equal to $x$. It completely describes the probability distribution of a random variable and serves as a fundamental tool for calculating probabilities and characterizing distributions. The function is defined for all real numbers and provides a unified way to handle both discrete and continuous variables, as well as mixed distributions.

Definition and mathematical formulation

Formally, for a random variable $X$ defined on a probability space $(\backslash Omega,\; \backslash mathcal\{F\},\; P)$, the CDF $F\_X:\; \backslash mathbb\{R\}\; \backslash rightarrow\; [0,1]$ is given by $F\_X(x)\; =\; P(X\; \backslash le\; x)$. This definition holds regardless of whether the distribution of $X$ is described by a probability mass function, a probability density function, or a more general measure. In the context of multivariate distributions, one defines a joint CDF, such as $F\_\{X,Y\}(x,y)\; =\; P(X\; \backslash le\; x,\; Y\; \backslash le\; y)$ for random variables $X$ and $Y$. The CDF is also directly related to the survival function, which is defined as $S(x)\; =\; P(X\; >\; x)\; =\; 1\; -\; F\_X(x)$.

Properties and characteristics

Every CDF is a monotonically non-decreasing and right-continuous function. As $x$ approaches negative infinity, the limit of the CDF is 0, and as it approaches positive infinity, the limit is 1. These properties are encapsulated in the fact that a CDF is a càdlàg function. Points where the CDF "jumps" correspond to atoms of the distribution, indicating a non-zero probability mass at specific values, which is characteristic of discrete distributions like the binomial distribution or Poisson distribution. For absolutely continuous distributions, the CDF is an absolutely continuous function and can be expressed as the integral of its probability density function.

Applications in probability and statistics

The CDF is instrumental in calculating probabilities for intervals, since . It is the primary function used in the probability integral transform, which states that if $F\_X$ is continuous, then the random variable $U\; =\; F\_X(X)$ follows a standard uniform distribution. This transform is the basis for inverse transform sampling, a fundamental method in Monte Carlo simulation. In statistical inference, the empirical distribution function, which is a step function based on a sample, serves as a non-parametric estimator of the true CDF and is central to methods like the Kolmogorov–Smirnov test. The CDF is also crucial in defining quantiles and percentiles, which are used in fields from finance to engineering.

Relationship to other functions

The CDF is intimately connected to other key functions in probability. For continuous distributions, it is the antiderivative (indefinite integral) of the probability density function, when the PDF exists. Conversely, the PDF is the derivative of the CDF at points where the derivative exists. For discrete distributions, the CDF is related to the probability mass function through a summation. The moment-generating function and characteristic function can be expressed as integrals involving the CDF. Furthermore, the hazard function used in survival analysis is defined as the ratio of the PDF to the survival function, which is derived from the CDF.

Examples and special cases

For the standard normal distribution, the CDF, often denoted $\backslash Phi(z)$, is a non-elementary integral but is widely tabulated and available in software like R and MATLAB. The CDF of the exponential distribution is $F(x)\; =\; 1\; -\; e^\{-\backslash lambda\; x\}$ for $x\; \backslash ge\; 0$. For a discrete example, the CDF of a binomial distribution with parameters $n$ and $p$ is a step function given by a sum of binomial probabilities. The CDF of the Cauchy distribution illustrates a distribution with "heavy tails" and no defined mean. In the case of a mixed distribution, such as one combining a point mass and a continuous part, the CDF will have both jumps and continuous increasing segments. Category:Probability theory Category:Statistical theory Category:Probability distributions