Content deleted Content added
m occuring -> occurring |
Rewrite and expand. |
||
Line 1:
In [[probability theory]], a '''probability mass function''' (abbreviated '''pmf''') gives the probability that a [[discrete random variable]] is exactly equal to some value. The probability mass function differs from the [[probability density function]] in that the latter, defined only for [[continuous random variable]]s, does not describe an actual probability but rather a rate of change in the [[cumulative distribution function]].
==Mathematical description==
Suppose that ''X'' is a discrete random variable, that is, that it takes values on some [[countable]] [[state space]] ''S''. We may assume that ''S'' ⊂ '''R'''. Then the probability mass function ''f''<sub>''X''</sub>(''x'') for ''X'' is given by
:<math>f_X(x) = \begin{cases}\mathrm{Pr}(X = x), &x\in S,\\0, &x\in \mathbb{R}\backslash S.\end{cases}</math>
Note that this explicitly defines ''f''<sub>''X''</sub>(''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 distribution|Bernoulli random variable]]), and hence the probability mass function is
:<math>f_X(x) = \begin{cases}0.5, &x \in \{0, 1\},\\0, &x \in \mathbb{R}\backslash\{0, 1\}.\end{cases}</math>
Probability mass functions may also be defined for any discrete random variable, including [[constant random variable|constant]], [[Binomial distribution|binomial]] (including [[Bernoulli distribution|Bernoulli]]), [[negative binomial distribution|negative binomial]], [[Poisson distribution|Poisson]], [[geometric distribution|geometric]] and [[hypergeometric distribution|hypergeometric]] random variables.
<!-- OLD TEXT
A probability function where the ___domain is a discrete random variable, and the range is the probability of that particular random variable occurring. For example, flip 3 coins and let X represent the number of heads. X = {0, 1, 2, 3}. The probability that X = x is (3Cx)(0.5^3)
Compare with: [[probability density function]].
-->
| |||