Search results
Results from the WOW.Com Content Network
Given any random variables X 1, X 2, ..., X n, the order statistics X (1), X (2), ..., X (n) are also random variables, defined by sorting the values (realizations) of X 1, ..., X n in increasing order. When the random variables X 1, X 2, ..., X n form a sample they are independent and identically distributed. This is the case treated below.
Zipf's law (/ z ɪ f /; German pronunciation:) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the n-th entry is often approximately inversely proportional to n. The best known instance of Zipf's law applies to the frequency table of words in a text or corpus of natural language:
Similar to convex order, Laplace transform order is established by comparing the expectation of a function of the random variable where the function is from a special class: () = (). This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with α {\displaystyle ...
This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement ...
Plot with random data showing heteroscedasticity: The variance of the y-values of the dots increases with increasing values of x. In statistics, a sequence of random variables is homoscedastic (/ ˌ h oʊ m oʊ s k ə ˈ d æ s t ɪ k /) if all its random variables have the same finite variance; this is also known as homogeneity of variance ...
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
This might be "random" on the scale of the entire sequence, but in a smaller block it would not be "random" (it would not pass their tests), and would be useless for a number of statistical applications. As random number sets became more and more common, more tests, of increasing sophistication were used.
In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function of that variable, which is defined as the expected value of , as a function of the real variable (the frequency parameter of the Fourier transform).