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# probability distribution function

# 1 Definition

Let $(\Omega,\mathfrak{B},\mu)$ be a measure space. A probability distribution function on $\Omega$ is a function $f:\Omega\longrightarrow\mathbb{R}$ such that:

1. $f$ is $\mu$-measurable

2. $f$ is nonnegative $\mu$-almost everywhere.

3.

The main feature of a probability distribution function is that it induces a probability measure $P$ on the measure space $(\Omega,\mathfrak{B})$, given by

$P(A):=\int_{A}f(x)\ d\mu=\int_{{\Omega}}1_{A}f(x)\ d\mu,$ |

for all $A\in\mathfrak{B}$. 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 $(\Omega,\mathfrak{B})$.

# 2 Examples

# 2.1 Discrete case

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 $f:I\longrightarrow\mathbb{R}$ satisfying the equation

$\sum_{{i\in I}}f(i)=1.$ |

One example is the Poisson distribution $P_{r}$ on $\mathbb{N}$ (for any real number $r$), which is given by

$P_{r}(i):=e^{{-r}}\frac{r^{i}}{i!}$ |

for any $i\in\mathbb{N}$.

Given any probability space $(\Omega,\mathfrak{B},\mu)$ 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 $I$.

# 2.2 Continuous case

Suppose $(\Omega,\mathfrak{B},\mu)$ equals $(\mathbb{R},\mathfrak{B}_{\lambda},\lambda)$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f:\mathbb{R}\longrightarrow\mathbb{R}$ 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 $x\in\mathbb{R}$. A well known example of a probability distribution function on $\mathbb{R}$ is the Gaussian distribution, or normal distribution

$f(x):=\frac{1}{\sigma\sqrt{2\pi}}e^{{-(x-m)^{2}/2\sigma^{2}}}.$ |

## Mathematics Subject Classification

60E99*no label found*

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