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A *distribution function* $F_X(x)$ (*Функция распределения*)

- is a function that defines the Probability of a Random Variable $X$ having values less than $x$
- i.e. $F_X(x) = P(X < x)$
- this, $F_X(x)$ defines the probability of $X$ taking value on the left of $x$
- for continuous - the area under the Probability Density Function from $-\infty$ to $x$
- it describes the Distribution of $X$

$0 \leqslant F(x) \leqslant 1$

$F_X(x)$ - monotonic non-increasing function (неубывающая)

- $\Rightarrow$ $P(a \leqslant X \leqslant b) = F_X(a) - F_X(b)$

if $\text{Dom}(X) = (a, b)$ (continuous)

- $F_X(x) = 0$ when $x < a$
- $F_X(x) = 1$ when $x \geqslant b$

A *density* of $F_X(x)$ is a function $f_X(x)$ s.t.

- $f_X(x) = F_X'(x)$

The probability that $X$ will take some value from interval $(a, b)$

- $P(a < X < b) = \int_a^b f_X(x) dx$

The cumulate distribution function $F_X(x)$ can be found by taking an integral of $f_X(x)$

- $F_X(x) = \int_{-\infty}^x f_X(x) dx$

- Гмурман В.Е., Теория вероятностей и математическая статистика -- 9-е издание. М.: Высш. шк., 2003.
- http://en.wikipedia.org/wiki/Probability_distribution
- http://en.wikipedia.org/wiki/Cumulative_distribution_function
- http://en.wikipedia.org/wiki/Probability_density_function