# Discrete Probability Variable - Distribution Function

## Probability Variable ξ

If we assign a numeric value to the elements of the event space, the resulting variable (random elementary events) is called a random variable (random, stochastic).

Example : Let's roll two dice! Add up the thrown numbers!
Event spacer:
possible outcomes, altogether 6x6=36:
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6},
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Probability variable ξ:
is the sum of the numbers thrown by the cube pair, possible values:
ξ ϵ{2,3,4,5,6,7,8,9,10,11,12}

## Distribution Function

The distribution function F(x) , also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate x takes on a value less than or equal to a number ξ .

$F\left(x\right)=P\left(\xi
Corresponding events for the values of probability variable ξ are :
ξ=2: {(1,1)}, altogether 1 case
ξ=3: {(1,2),(2,1)}, altogether 2 case
ξ=4: {(1,3),(3,1),(2,2)}, altogether 3 case
ξ=5: {(1,4),(4,1),(2,3),(2,3)}, altogether 4 case
ξ=6: {(1,5),(5,1),(2,4),(4,2), (3,3)}, altogether 5 case
ξ=7: {(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)}, altogether 6 case
ξ=8: {(2,6),(6,2),(3,5),(5,3), (4,4)}, altogether 5 case
ξ=9: {(3,6),(6,3),(4,5),(5,4)}, altogether 4 case
ξ=10: {(4,6),(6,4),(5,5)}, altogether 3 case
ξ=11: {(5,6),(6,5)}, altogether 2 case
ξ=12: {(6,6)}, altogether 1 case

Values of distribution function F (x) for different values of x :

$F\left(x=1\right)=P\left(\xi <1\right)=0$
$F\left(x=2\right)=P\left(\xi <2\right)=0$
$F\left(x=3\right)=P\left(\xi <3\right)=\frac{1}{36}=0.03$
$F\left(x=4\right)=P\left(\xi <4\right)=\frac{1}{36}+\frac{2}{36}=\frac{3}{36}=0.08$
$F\left(x=5\right)=P\left(\xi <5\right)=\frac{1}{36}+\frac{2}{36}+\frac{3}{36}=\frac{6}{36}=0.17$
$F\left(x=6\right)=P\left(\xi <6\right)=0,28$
$F\left(x=7\right)=P\left(\xi <7\right)=0,42$
$F\left(x=8\right)=P\left(\xi <8\right)=0,58$
$F\left(x=9\right)=P\left(\xi <9\right)=0,72$
$F\left(x=10\right)=P\left(\xi <10\right)=0,83$
$F\left(x=11\right)=P\left(\xi <11\right)=0,92$
$F\left(x=12\right)=P\left(\xi <12\right)=0,97$
$F\left(x=13\right)=P\left(\xi <13\right)=1$

Keywords: Discrete Probability Variable - Distribution Function