The distribution function

The distribution function of a random variable $X$ is a function that assigns, for every point, the probability accumulated up to the above mentioned value. That is:

$F (X) = p (X \leq x)$

Example

For example, we will compute the distribution function of the random variable $X$ , resulting from throwing a perfect dice.

The following table shows the values of $F (x)$ :

x	F (x)
$x < 1$	$0$
$1 \leq x < 2$	$\frac{1}{6}$
$1 \leq x < 3$	$\frac{2}{6}$
$1 \leq x < 4$	$\frac{3}{6}$
$1 \leq x < 5$	$\frac{4}{6}$
$1 \leq x < 6$	$\frac{5}{6}$
$x \leq 6$	$1$

The value of the function distribution in $- \infty$ will always be $0$ , while the value in $+ \infty$ will always be $1$ .

This turns out to be quite intuitive, since the probability that the value of $x$ is smaller than $- \infty$ is zero, and the probability that it is less than $+ \infty$ is $1$ (since it is always less or equal to $+ \infty$ ).

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Since it is a discrete random variable, the distribution function will be discrete. Note that the probability of obtaining a result lower than $5, 2$ is the same as that of lower than $5, 3$ or $5, 9$ .