{"history":[],"article":{"id":"018ac920-f731-4bda-b7d3-032cd52891b9","title":"Cumulative distribution function","slug":"cumulative-distribution-function","content":":::summary\n:::\n\n# \nIn [probability theory](https://bliptext.com/articles/[probability](https://bliptext.com/articles/probability)-theory) and [statistics](https://bliptext.com/articles/statistics), the **cumulative **distribution function**** (**CDF**) of a real-valued [random variable](https://bliptext.com/articles/random-variable) X, or just **distribution function** of X, evaluated at x, is the [probability](https://bliptext.com/articles/probability) that X will take a value less than or equal to x.\nEvery [probability distribution](https://bliptext.com/articles/probability-distribution) [supported](https://bliptext.com/articles/support-measure-theory) on the real numbers, discrete or \"mixed\" as well as [continuous](https://bliptext.com/articles/continuous-variable), is uniquely identified by a right-[continuous](https://bliptext.com/articles/continuous-variable) [monotone increasing](https://bliptext.com/articles/monotonic-function) function (a [càdlàg](https://bliptext.com/articles/c-dl-g) function) satisfying and.\nIn the case of a scalar [continuous distribution](https://bliptext.com/articles/continuous-distribution), it gives the area under the [probability density function](https://bliptext.com/articles/probability-density-function) from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of [multivariate random variables](https://bliptext.com/articles/multivariate-random-variable).\n# [Cumulative distribution function for the exponential distribution | upload.wikimedia.org/wikipedia/commons/8/80/Exponential///distribution///cdf.svg]\n# [Cumulative distribution function for the normal distribution | upload.wikimedia.org/wikipedia/commons/c/ca/Normal///Distribution///CDF.svg]\n\n# Definition\nThe cumulative distribution function of a real-valued [random variable](https://bliptext.com/articles/random-variable) X is the function given by\nwhere the right-hand side represents the [probability](https://bliptext.com/articles/probability) that the random variable X takes on a value less than or equal to x.\nThe probability that X lies in the semi-closed [interval](https://bliptext.com/articles/interval-mathematics) (a,b], where a < b, is therefore\nIn the definition above, the \"less than or equal to\" sign, \"≤\", is a convention, not a universally used one (e.g. Hungarian literature uses \"<\"), but the distinction is important for discrete distributions.\nThe proper use of tables of the [binomial](https://bliptext.com/articles/binomial-distribution) and [Poisson distributions](https://bliptext.com/articles/poisson-distribution) depends upon this convention.\nMoreover, important formulas like [Paul Lévy](https://bliptext.com/articles/paul-l-vy-mathematician)'s inversion formula for the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) also rely on the \"less than or equal\" formulation.\nIf treating several random variables etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted.\nIt is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for [probability density functions](https://bliptext.com/articles/probability-density-function) and [probability mass functions](https://bliptext.com/articles/probability-mass-function).\nThis applies when discussing general distributions: some specific distributions have their own conventional notation, for example the [normal distribution](https://bliptext.com/articles/normal-distribution) uses \\Phi and \\phi instead of F and f, respectively.\nThe probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating using the [Fundamental Theorem of Calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus); i.e. given F(x),\nas long as the derivative exists.\nThe CDF of a [continuous random variable](https://bliptext.com/articles/continuous-random-variable) X can be expressed as the integral of its probability density function f_X as follows:\nIn the case of a random variable X which has distribution having a discrete component at a value b,\nIf F_X is continuous at b, this equals zero and there is no discrete component at b.\n\n# Properties\nEvery cumulative distribution function F_X is [non-decreasing](https://bliptext.com/articles/monotone-increasing) and [right-continuous](https://bliptext.com/articles/right-continuous), which makes it a [càdlàg](https://bliptext.com/articles/c-dl-g) function.\nFurthermore,\nEvery function with these three properties is a CDF, i.e., for every such function, a [random variable](https://bliptext.com/articles/random-variable) can be defined such that the function is the cumulative distribution function of that [random variable](https://bliptext.com/articles/random-variable).\nIf X is a purely [discrete random variable](https://bliptext.com/articles/discrete-random-variable), then it attains values with probability, and the CDF of X will be [discontinuous](https://bliptext.com/articles/discontinuity-mathematics) at the points x_i:\nIf the CDF F_X of a real valued random variable X is [continuous](https://bliptext.com/articles/continuous-function), then X is a [continuous](https://bliptext.com/articles/continuous-function) random variable; if furthermore F_X is absolutely [continuous](https://bliptext.com/articles/continuous-function), then there exists a [Lebesgue-integrable](https://bliptext.com/articles/lebesgue-integral) function f_X(x) such that\nfor all real numbers a and b. The function f_X is equal to the [derivative](https://bliptext.com/articles/derivative) of F_X [almost everywhere](https://bliptext.com/articles/almost-everywhere), and it is called the [probability density function](https://bliptext.com/articles/probability-density-function) of the distribution of X.\nIf X has finite [L1-norm](https://bliptext.com/articles/l1-norm), that is, the expectation of |X| is finite, then the expectation is given by the [Riemann–Stieltjes integral](https://bliptext.com/articles/riemann-stieltjes-integral)\nand for any x \\geq 0,\nas well as\nas shown in the diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of F_X).\nIn particular, we have\nIn addition, the (finite) expected value of the real-valued random variable X can be defined on the graph of its cumulative distribution function as illustrated by the [drawing](https://bliptext.com/articles/expected-value) in the [definition of expected value for arbitrary real-valued random variables](https://bliptext.com/articles/expected-value).\n# [From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part. | upload.wikimedia.org/wikipedia/commons/2/2e/Discrete///probability///distribution///illustration.svg]\n# [Example of a cumulative distribution function with a countably infinite set of discontinuities. | upload.wikimedia.org/wikipedia/commons/e/ea/Discrete///probability///distribution///with///a///countable///set///of///discontinuities.svg]\n# [CDF plot with two red rectangles, illustrating two inequalities | upload.wikimedia.org/wikipedia/commons/0/08/CDF///plot///with///two///red///rectangles%2C///illustrating///%28-x%29F%28x%29///and///x%281-F%28x%29%29.png]\n\n# Examples\nAs an example, suppose X is [uniformly distributed](https://bliptext.com/articles/uniform-distribution-continuous) on the unit interval [0,1].\nThen the CDF of X is given by\nSuppose instead that X takes only the discrete values 0 and 1, with equal probability.\nThen the CDF of X is given by\nSuppose X is [exponential distributed](https://bliptext.com/articles/exponential-distribution).\nThen the CDF of X is given by\nHere λ > 0 is the parameter of the distribution, often called the rate parameter.\nSuppose X is [normal distributed](https://bliptext.com/articles/normal-distribution).\nThen the CDF of X is given by\nHere the parameter \\mu is the [mean](https://bliptext.com/articles/mean) or expectation of the distribution; and \\sigma is its standard deviation.\nA table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the [standard normal table](https://bliptext.com/articles/standard-normal-table), the **unit normal table**, or the **Z table**.\nSuppose X is [binomial distributed](https://bliptext.com/articles/binomial-distribution).\nThen the CDF of X is given by\nHere p is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n independent experiments, and is the \"floor\" under k, i.e. the [greatest integer](https://bliptext.com/articles/greatest-integer) less than or equal to k.\n\n# Derived functions\n\n## Complementary cumulative distribution function (tail distribution)\nSometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level.\nThis is called the **' (**') or simply the **' or **', and is defined as\nThis has applications in [statistical](https://bliptext.com/articles/statistics) [hypothesis testing](https://bliptext.com/articles/hypothesis-test), for example, because the one-sided [p-value](https://bliptext.com/articles/p-value) is the probability of observing a test statistic at least as extreme as the one observed.\nThus, provided that the [test statistic](https://bliptext.com/articles/test-statistic), T, has a continuous distribution, the one-sided [p-value](https://bliptext.com/articles/p-value) is simply given by the ccdf: for an observed value t of the [test statistic](https://bliptext.com/articles/test-statistic)\nIn [survival analysis](https://bliptext.com/articles/survival-analysis), \\bar F_X(x) is called the [survival function](https://bliptext.com/articles/survival-function) and denoted S(x), while the term reliability function is common in [engineering](https://bliptext.com/articles/engineering).\nProof:\nAssuming X has a density function f_X, for any c > 0 Then, on recognizing and rearranging terms, as claimed.\nIf the random variable can only take non-negative integer values, this is equivalent to\n\n## Folded cumulative distribution\nWhile the plot of a cumulative distribution F often has an S-like shape, an alternative illustration is the **folded cumulative distribution** or **mountain plot**, which folds the top half of the graph over, that is where 1_{\\{A\\}} denotes the [indicator function](https://bliptext.com/articles/indicator-function) and the second summand is the [survivor function](https://bliptext.com/articles/survivor-function), thus using two scales, one for the upslope and another for the downslope.\nThis form of illustration emphasises the [median](https://bliptext.com/articles/median-statistics), [dispersion](https://bliptext.com/articles/dispersion-statistics) (specifically, the [mean absolute deviation](https://bliptext.com/articles/mean-absolute-deviation) from the [median](https://bliptext.com/articles/median-statistics) ) and [skewness](https://bliptext.com/articles/skewness) of the distribution or of the empirical results.\n# [Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1. | upload.wikimedia.org/wikipedia/commons/b/b5/Folded-cumulative-distribution-function.svg]\n\n## Inverse distribution function (quantile function)\nIf the CDF F is strictly increasing and continuous then is the unique real number x such that F(x) = p. This defines the **inverse distribution function** or [quantile function](https://bliptext.com/articles/quantile-function).\nSome distributions do not have a unique inverse (for example if f_X(x)=0 for all a 0 Then, on recognizing and rearranging terms, as claimed.\nIf the random variable can only take non-negative integer values, this is equivalent to\n\n## Folded cumulative distribution\nWhile the plot of a cumulative distribution F often has an S-like shape, an alternative illustration is the **folded cumulative distribution** or **mountain plot**, which folds the top half of the graph over, that is where 1_{\\{A\\}} denotes the [indicator function](https://bliptext.com/articles/indicator-function) and the second summand is the [survivor function](https://bliptext.com/articles/survivor-function), thus using two scales, one for the upslope and another for the downslope.\nThis form of illustration emphasises the [median](https://bliptext.com/articles/median-statistics), [dispersion](https://bliptext.com/articles/dispersion-statistics) (specifically, the [mean absolute deviation](https://bliptext.com/articles/mean-absolute-deviation) from the [median](https://bliptext.com/articles/median-statistics) ) and [skewness](https://bliptext.com/articles/skewness) of the distribution or of the empirical results.\n# [Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1. | upload.wikimedia.org/wikipedia/commons/b/b5/Folded-cumulative-distribution-function.svg]\n\n## Inverse distribution function (quantile function)\nIf the CDF F is strictly increasing and continuous then is the unique real number x such that F(x) = p. This defines the **inverse distribution function** or [quantile function](https://bliptext.com/articles/quantile-function).\nSome distributions do not have a unique inverse (for example if f_X(x)=0 for all a

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