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The geometric distribution defined on ... Substituting this estimate in the formula for the expected value of a geometric distribution and solving for ...
Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as: + (=) = [1] Note that for M = 0 or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation.
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure.
This function can be considered as a generalization of the geometric series. The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function M ( a , c , z ) = lim b → ∞ 2 F 1 ( a , b ; c ; b − 1 z ) {\displaystyle M(a,c,z)=\lim _{b\to \infty }{}_{2}F_{1}(a,b;c;b^{-1}z)}
There is a one-to-one correspondence between cumulative distribution functions and characteristic functions, so it is possible to find one of these functions if we know the other. The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f).
Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability. ( Buffon's needle ) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?
In probability theory, the matrix geometric method is a method for the analysis of quasi-birth–death processes, continuous-time Markov chain whose transition rate matrices with a repetitive block structure. [1] The method was developed "largely by Marcel F. Neuts and his students starting around 1975." [2]
The Riemann zeta function ζ(s) is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number s with real part greater than 1 by the following formula: ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty ...