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probability distribution function
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(Definition)
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Let $(\Omega, \borel, \mu)$ be a measure space. A probability distribution function on $\Omega$ is a function $f: \Omega \longrightarrow \reals$ such that:
- $f$ is $\mu$ -measurable
- $f$ is nonnegative $\mu$ -almost everywhere.
- $f$ satisfies the equation $$ \int_{\Omega} f(x)\ d\mu = 1 $$
The main feature of a probability distribution function is that it induces a probability measure $P$ on the measure space $(\Omega, \borel)$ , given by $$ P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu, $$ for all $X \in \borel$ . The measure $P$ is called the associated probability measure of $f$ . Note that $P$ and $\mu$ are different measures, even though they both share the same underlying measurable space $(\Omega, \borel)$ .
Let $I$ be a countable set, and impose the counting measure on $I$ ($\mu(A) := |A|$ , the cardinality of $A$ , for any subset $A \subset I$ ). A probability distribution function on $I$ is then a nonnegative function $f: I \longrightarrow \reals$ satisfying the equation $$ \sum_{i \in I} f(i) = 1. $$
One example is the Poisson distribution $P_r$ on $\naturals$ (for any real number $r$ ), which is given by $$ P_r(i) := e^{-r} \frac{r^i}{i!} $$ for any $i \in \naturals$ .
Given any probability space $(\Omega, \borel, \mu)$ and any random variable $X: \Omega \longrightarrow I$ , we can form a distribution function on $I$ by taking $f(i) := \mu(\{X = i\})$ . The resulting function is called the distribution of $X$ on $I$ .
Suppose $(\Omega, \borel, \mu)$ equals $(\reals, \borel_\lambda, \lambda)$ , the real numbers equipped with Lebesgue measure. Then a probability distribution function $f: \reals \longrightarrow \reals$ is simply a measurable, nonnegative almost everywhere function such that $$ \int_{-\infty}^\infty f(x)\ dx = 1. $$ The associated measure has Radon-Nikodym derivative with respect to $\lambda$ equal to $f$ : $$
\frac{dP}{d\lambda} = f. $$ One defines the cumulative distribution function $F$ of $f$ by the formula $$ F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt, $$ for all $x \in \reals$ . A well known example of a probability distribution function on $\reals$ is the Gaussian distribution, or normal distribution $$ f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2}. $$
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Cross-references: Gaussian, formula, almost everywhere, measurable, Lebesgue measure, random variable, probability space, real number, Poisson distribution, subset, cardinality, counting measure, countable, measurable space, measure, probability measure, induces, equation, function, measure space
There are 62 references to this entry.
This is version 7 of probability distribution function, born on 2002-04-29, modified 2007-09-22.
Object id is 2884, canonical name is Distribution.
Accessed 57148 times total.
Classification:
| AMS MSC: | 60E99 (Probability theory and stochastic processes :: Distribution theory :: Miscellaneous) |
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Pending Errata and Addenda
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