(Cumulative) Distribution Function

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


Properties

Property 1: Sum of Probabilities

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

Property 2: Monotonicity

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

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

Property 3

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

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


Probability Density Function

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 $f_X(x)$

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


Sources

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