Cumulative distribution function
Function giving probability that a variable is ≤ a value.
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, often simply called the distribution function, gives the probability that X will take a value less than or equal to a given point x. This function is defined for all real numbers and is denoted by F(x) = P(X ≤ x). The probability that X falls within a semi-closed interval (a, b], where a < b, is then F(b) − F(a). While the "less than or equal to" convention is standard, some sources, such as Hungarian literature, may use a strict inequality.
Every probability distribution on the real numbers—whether continuous, discrete, or mixed—is uniquely characterized by its CDF, which is a right-continuous, monotone increasing function (a càdlàg function) with limits of 0 as x approaches negative infinity and 1 as x approaches positive infinity. For a continuous distribution, the CDF represents the area under the probability density function from negative infinity to x. The CDF can also be used to define distributions for multivariate random variables.
The complementary cumulative distribution function (ccdf), also called the tail distribution or survival function, is defined as F̄(x) = 1 − F(x) and answers how often the variable exceeds a given level. This is used in statistical hypothesis testing for p-values and in survival analysis, where it is known as the survival function S(t). For non-negative random variables with an expectation, Markov's inequality provides a bound: P(X ≥ a) ≤ E(X)/a. The expected value of a non-negative random variable can be expressed in terms of its CDF as E(X) = ∫₀^∞ (1 − F(x)) dx, with a discrete analogue for integer-valued variables.
If the CDF is strictly increasing and continuous, it has an inverse called the quantile function, Q(p) = F⁻¹(p), which gives the unique x such that F(x) = p. For distributions where F is constant over intervals, a generalized inverse is defined. The inverse is used in random number generation via inverse transform sampling. The empirical distribution function, based on a sample, estimates the true CDF and converges to it with probability 1.
- field
- Probability theory and statistics
- known_for
- Defining the probability that a random variable takes a value less than or equal to a given point; uniquely identifying probability distributions via a right-continuous monotone increasing function
Lore & Background
The cumulative distribution function (CDF) of a real-valued random variable X, denoted F_X(x), is defined as the probability that X takes a value less than or equal to x. For any real numbers a and b with a < b, the probability that X lies in the interval (a, b] is given by F_X(b) − F_X(a). Every probability distribution on the real numbers, whether continuous, discrete, or mixed, is uniquely identified by a function that is right-continuous, monotone increasing (a càdlàg function), and satisfies limits of 0 at negative infinity and 1 at positive infinity. For a scalar continuous distribution, the CDF represents the area under the probability density function from negative infinity to x. The CDF is also used to specify multivariate distributions. A key property is that the CDF can determine the expected value of X through the area of the region above the CDF and below the line y=1. The complementary cumulative distribution function, or tail distribution, defined as F̄(x) = 1 − F(x), gives the probability that X exceeds a particular level; in survival analysis, this is called the survival function. The folded cumulative distribution, or mountain plot, folds the top half of the CDF graph over to emphasize the median, dispersion, and skewness. The inverse distribution function, or quantile function, is defined for a strictly increasing and continuous CDF as the unique real number x such that F(x) = p; for non-unique inverses, a generalized inverse is used, defined as the infimum of x such that F(x) ≥ p. The inverse CDF is nondecreasing and is used in random number generation via inverse transform sampling. The empirical distribution function estimates the underlying CDF from a sample and converges with probability 1 to that distribution.
Reader's Guide
The cumulative distribution function is a fundamental tool in probability theory and statistics, as it uniquely identifies any probability distribution on the real numbers, whether continuous, discrete, or mixed. For a continuous random variable, the CDF gives the area under the probability density function from negative infinity to x, and the density can be recovered by differentiating the CDF where the derivative exists. For a discrete random variable, the CDF is a step function that jumps at the values the variable can take, with the jump height equal to the probability of that value. The CDF is always non-decreasing and right-continuous (a càdlàg function), with limits 0 as x → −∞ and 1 as x → +∞. Conversely, any function with these three properties is a CDF for some random variable. The CDF is also used to specify the distribution of multivariate random variables. Notation conventions include using a capital F for the CDF, with specific distributions sometimes using other letters (e.g., Φ for the normal distribution).
Did You Know?
- The cumulative distribution function is right-continuous and non-decreasing, making it a càdlàg function.
- For a continuous random variable, the probability density function can be obtained by differentiating the CDF using the Fundamental Theorem of Calculus.
- The probability that a random variable X equals a specific value b is given by F_X(b) − lim_{x→b−} F_X(x).
- Every function that is non-decreasing, right-continuous, and has limits 0 at −∞ and 1 at +∞ is the CDF of some random variable.
More in Probability And Stochastic Processes 1-21
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record