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Power Series Distribution

A discrete random variable X is said to have a generalised power series distribution if its probability function is given by,

$f\left&space;(&space;x;a,\theta&space;\right&space;)=\frac{a_{x}\theta&space;^{x}}{f\left&space;(&space;\theta&space;\right&space;)};&space;x=0,1,2,3....;&space;a_{x}\geq&space;0.$

where f(θ) is a generating function and  f(θ) is positive finite and diferentiable. Power series distribution is a discrete probability distribution
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Properties of Power series distribution

Some special properties of power series distribution are given-
• If θ=p/(1-p), f(θ)=(1+θ)^n and s={1,2,3,…,n), a set of (n+1) non-negative integers then the power series distribution is tends to binomial distribution
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• If f(θ)=e^θ and s={0,1,2,3,…,∞} then the distribution tends to poisson distribution.
• If θ=p/(1-p), f(θ)=(1+θ)^-n and s={0,1,2,3,…,∞), then the power series distribution tends to negative binomial distribution.
• If f(θ)=-log(1-θ) and s={1,2,….}, then the power series distribution tends to logarthmic distribution.

Releated

Probability Distributions in Statistics

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Spread the love          In statistics, probability distributions are very important measure which related to the random variable and statistical data. Every data patern follows a probability distribution. Continuous data follow the continuous distribution and discrete data follow the discrete distributions. According to wikipedia, “In probability theory and statistics, a probability distribution is the mathematical function that […]

Gamma Distribution definition, formula and applications

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Spread the love          In probabilistic statistics, the gamma distribution is a two-parameter family of continuous probability distributions which is widely used in different sectors. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of it. Gamma Distribution A continuous random variable X is said to have a generalised gamma distribution with parameter α and […]