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. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. 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. 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. 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. Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value.
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. In probability theory and statistics, the cumulative distribution function ( CDF) of a real-valued random variable X. The distribution is obtained by performing a number of Bernoulli trials. Cumulative distribution function for the normal distribution Binomial distribution is a probability distribution that summarises the likelihood that a variable will take one of two independent values under a given set of parameters.
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