Cumulative distribution function

Lore & Background

The cumulative distribution function is defined for a real-valued random variable X as F_X(x) = P(X ≤ x). The right-hand side represents the probability that X takes on a value less than or equal to x. The probability that X lies in the semi-closed interval (a, b], where a < b, is therefore F_X(b) − F_X(a). The use of 'less than or equal to' is a convention, not universally used (e.g., Hungarian literature uses '<'), but the distinction is important for discrete distributions, and formulas like Paul Lévy's inversion formula rely on the 'less than or equal' formulation.

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.