Poisson Binomial Mass Function

A Poisson Binomial Mass Function is a Binomial Mass Function for a Poisson Binomial Random Variable.

• AKA: Poisson Binomial.
• Context:
  • It can be instantiated in a Poisson Binomial Distribution.
  • …
• Counter-Example(s):
  • Negative Binomial Mass Function.
  • Binomial Mass Function.
• See: Controlled Experiment Evaluation Task.

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References

2011

• http://en.wikipedia.org/wiki/Poisson_binomial_distribution
  • QUOTE: In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials. In other words, it is the probability distribution of the number of successes in a sequence of [math]\displaystyle{ n }[/math] independent yes/no experiments with success probabilities [math]\displaystyle{ p_1, p_2, \dots, p_n }[/math]. The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is [math]\displaystyle{ p_1 = p_2 = \dots = p_n }[/math] .
  • QUOTE: Since a Poisson binomial distributed variable is a sum of [math]\displaystyle{ n }[/math] independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the [math]\displaystyle{ n }[/math] Bernoulli distributions:
    [math]\displaystyle{ \mu = \sum\limits_{i=1}^n p_i }[/math]
    [math]\displaystyle{ \sigma^2 =\sum\limits_{i=1}^n (1-p_i) p_i }[/math]
  • QUOTE: The probability of having [math]\displaystyle{ k }[/math] successful trials out of a total of [math]\displaystyle{ n }[/math] can be written as the sum
    [math]\displaystyle{ Pr(K=k) = \sum\limits_{A\in {{F}_{k\lt /code\gt {\prod\limits_{i\in A}\lt code\gt p}_{i\lt /code\gt \prod\limits_{j\in {{A}^{c\lt /code\gt {(1-{{p}_{j}})}} }[/math]
    where [math]\displaystyle{ F_k }[/math] is the set of all subsets of [math]\displaystyle{ k }[/math] integers that can be selected from {1,2,3,...,n}. For example, if n = 3, then
    [math]\displaystyle{ {{F}_{2}}=\left\{ \{1,2\},\{1,3\},\{2,3\} \right\} }[/math].
    [math]\displaystyle{ A^c }[/math] is the complement of [math]\displaystyle{ A }[/math], i.e. [math]\displaystyle{ A^c =\{1,2,3,\dots,n\}\backslash A }[/math].

2002

• QuickCalcs Online Calculator: http://www.graphpad.com/quickcalcs/probability1.cfm
  • QUOTE: The Poisson distribution applies when you are counting the number of objects in a certain volume or the number of events in a certain time period. You know the average number of counts, and wish to know the chance of actually observing various numbers of objects or events.

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