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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.
A variable is considered dependent if it depends on an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function), on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of ...
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
The dependent variable could be ranked on the following list: complete cure, improved symptoms, no change, worsened symptoms, or death. [ citation needed ] Another example application are Likert-type items commonly employed in survey research, where respondents rate their agreement on an ordered scale (e.g., "Strongly disagree" to "Strongly ...
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 ...
The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2. The binomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success.
Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is ...
Independent increments are a basic property of many stochastic processes and are often incorporated in their definition. The notion of independent increments and independent S-increments of random measures plays an important role in the characterization of Poisson point process and infinite divisibility.