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MATHEMATICAL DESCRIPTION Suppose that ''X'' is a discrete random variable, taking values on some Countable Sample Space ''S'' ⊆ R. Then the probability mass function ''f''''X''(''x'') for ''X'' is given by : Note that this explicitly defines ''f''''X''(''x'') for all Real Number s, including all values in R that ''X'' could never take; indeed, it assigns such values a probability of zero. (Alternatively, think of Pr(''X'' = ''x'') as 0 when ''x'' ∈ R\''S''.) The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where ''x'' ∈ R\''S'') the derivative is zero, just as the probability mass function is zero at all such points. EXAMPLES A simple example of a probability mass function is the following. Suppose that ''X'' is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that ''X'' = ''x'' is just 0.5 on the state space {0, 1} (this is a Bernoulli Random Variable ), and hence the probability mass function is : Probability mass functions may also be defined for any discrete random variable, including Constant , Binomial (including Bernoulli ), Negative Binomial , Poisson , Geometric and Hypergeometric random variables. |
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