Cumulative distribution function
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In probability theory and statistics, the cumulative distribution function (CDF), or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far" function of the probability distribution. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.
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[edit] Definition
For every real number x, the cumulative distribution function of a real-valued random variable X is given by
where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x. The probability that X lies in the interval (a, b], where a < b, is therefore
Here the notation (a, b], indicates a semi-closed interval.
If treating several random variables X, Y, ... etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution.
The CDF of a continuous random variable X can be defined in terms of its probability density function ƒ as follows:
Note that in the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation.
In the case of a random variable X which has distribution having a discrete component at a value x0,
where F(x0-) denotes the limit from the left of F at x0: i.e. lim F(y) as y increases towards x0.
[edit] Properties
Every cumulative distribution function F is (not necessarily strictly) monotone non-decreasing (see monotone increasing) and right-continuous. Furthermore,
Every function with these four properties is a CDF: more specifically, for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable according to the definition above.[citation needed]
The properties imply that all CDFs are càdlàg functions.[citation needed]
If X is a purely discrete random variable, then it attains values x1, x2, ... with probability pi = P(xi), and the CDF of X will be discontinuous at the points xi and constant in between:
If the CDF F of X is continuous, then X is a continuous random variable; if furthermore F is absolutely continuous, then there exists a Lebesgue-integrable function f(x) such that
for all real numbers a and b. The function f is equal to the derivative of F almost everywhere, and it is called the probability density function of the distribution of X.
[edit] Point probability
The "point probability" that X is exactly b can be found as
[edit] Examples
As an example, suppose X is uniformly distributed on the unit interval [0, 1]. Then the CDF of X is given by
Suppose instead that X takes only the discrete values 0 and 1, with equal probability. Then the CDF of X is given by
[edit] Derived functions
[edit] Complementary cumulative distribution function (tail distribution)
Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as
This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic, T, has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value t of the test statistic
In survival analysis, is called the survival function and denoted , while the term reliability function is common in engineering.
- Properties
- For a non-negative continuous random variable having an expectation, Markov's inequality states that[1]
- As , and in fact
- Proof:[citation needed] Assuming X has a density function f, for any
- Then, on recognizing and rearranging terms,
- as claimed.
[edit] Folded cumulative distribution
While the plot of a cumulative distribution 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,[2][3] thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median and dispersion (the mean absolute deviation from the median[4]) of the distribution or of the empirical results.
[edit] Inverse distribution function (quantile function)
If the CDF F is strictly increasing and continuous then is the unique real number such that . In such a case, this defines the inverse distribution function or quantile function.
Unfortunately, the distribution does not, in general, have an inverse. One may define, for , the generalized inverse distribution function:
- Example 1: The median is .
- Example 2: Put . Then we call the 95th percentile.
The inverse of the cdf is called the quantile function.
The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions. Some useful properties of the inverse cdf are:
- is nondecreasing
- if and only if
- If has a distribution then is distributed as . This is used in random number generation using the inverse transform sampling-method.
- If is a collection of independent -distributed random variables defined on the same sample space, then there exist random variables such that is distributed as and with probability 1 for all .
[edit] Multivariate case
When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X,Y, the joint CDF is given by
where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x and that Y takes on a value less than or equal to y.
Every multivariate CDF is:
- Monotonically non-decreasing for each of its variables
- Right-continuous for each of its variables.
- and
[edit] Use in statistical analysis
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various statistical hypothesis tests. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.
[edit] Kolmogorov–Smirnov and Kuiper's tests
The Kolmogorov–Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.
[edit] See also
[edit] References
- ^ Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1-58488-059-7 page 49.
- ^ Gentle, J.E. (2009). Computational Statistics. Springer. http://books.google.de/books?id=m4r-KVxpLsAC&lpg=PA348&ots=8Wxj0G_GC6&dq=folded%20cumulative%20distribution%20or%20mountain%20plot&hl=en&pg=PA348#v=onepage&q=folded%20cumulative%20distribution%20or%20mountain%20plot&f=false. Retrieved 2010-08-06.[page needed]
- ^ Monti, K.L. (1995). "Folded Empirical Distribution Function Curves (Mountain Plots)". The American Statistician 49: 342–345. JSTOR 2684570.
- ^ Xue, J. H.; Titterington, D. M. (2011). "The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile". Statistics & Probability Letters 81 (8): 1179–1182. doi:10.1016/j.spl.2011.03.014.<
[edit] External links
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