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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 test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test , test for divergence , or the divergence test . Ratio test
Many authors do not name this test or give it a shorter name. [2] When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.
(That is, the two dice are independent.) If, however, the 1st die's result is a 3, and someone tells you about a third event - that the sum of the two results is even - then this extra unit of information restricts the options for the 2nd result to an odd number. In other words, two events can be independent, but NOT conditionally independent. [2]
Then X 1 has the Bernoulli distribution with expected value μ = 0.5 and variance σ 2 = 0.25. The subsequent random variables X 2 , X 3 , ... will all be distributed binomially . As n grows larger, this distribution will gradually start to take shape more and more similar to the bell curve of the normal distribution.
Independent means that the sample items are all independent events. In other words, they are not connected to each other in any way; [ 2 ] knowledge of the value of one variable gives no information about the value of the other and vice versa.
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion z-test", for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic. [35]