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The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1. In probability and statistics, a probability mass function (sometimes called probability function or frequency function [1]) is a function that gives the probability that a discrete random variable is exactly equal to some value. [2]
The function (,) serves as a normalization constant so the probability mass function sums to one. Note that Z ( λ , ν ) {\displaystyle Z(\lambda ,\nu )} does not have a closed form. The domain of admissible parameters is λ , ν > 0 {\displaystyle \lambda ,\nu >0} , and 0 < λ < 1 {\displaystyle 0<\lambda <1} , ν = 0 {\displaystyle \nu =0} .
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
The Dirac comb of period 2 π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function. It represents a discrete probability distribution concentrated at 2 π n — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of based on a random sample is the sample mean. The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.
The form of the conjugate prior can generally be determined by inspection of the probability density or probability mass function of a distribution. For example, consider a random variable which consists of the number of successes in Bernoulli trials with unknown probability of success in [0,1].
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function. [10]
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.