Discrete triangular distribution

The discrete triangular distribution is a special probability distribution in stochastics . It is one of the discrete probability distributions on a finite set and is a univariate probability distribution .

This probability distribution is typically used for the sum of identical, uniformly distributed random variables, which results in a symmetrical triangular distribution.

Symmetrical triangular distribution

One chooses two with , so is selected as support , the amount ${\displaystyle a<b\in \mathbb {Z} }$ ${\displaystyle b-a=n-1}$

${\displaystyle T:=\{2a,2a+1,2a+2,\dotsc ,2b-1,2b\}}$

and defines the probability function for ${\displaystyle x\in \mathbb {Z} }$

${\displaystyle \operatorname {P} (X=x)=f(x)={\begin{cases}{\frac {x-2a+1}{n^{2}}}&{\text{für }}2a\leq x\leq a+b\\{\frac {2b-x+1}{n^{2}}}&{\text{für }}a+b\leq x\leq 2b\\0&{\text{sonst}}\end{cases}}}$

The expected value is , the variance is twice the variance of the random variable uniformly distributed to the amount: . ${\displaystyle a+b}$ ${\displaystyle \operatorname {Var} (X)={\frac {(b-a+2)(b-a)}{6}}}$ ${\displaystyle \{a,a+1,a+2,\dotsc ,b-1,b\}}$

Asymmetrical triangular distribution

The distribution on the crowd: with ${\displaystyle T:=\{a,a+1,a+2,\dotsc ,b-1,b\}}$

${\displaystyle \operatorname {P} (X=x)=f(x)={\begin{cases}{\frac {x-a+1}{n(n+1)/2}}&{\text{für }}a\leq x\leq b\\0&{\text{sonst}}\end{cases}}}$

is a discrete counterpart to the continuous triangular distribution, which is the amount of the difference between uniformly distributed random variables .

Examples

For example, in the board games Backgammon or Settlers of Catan , the sum of two dice is considered. This is symmetrically distributed in a triangle on the carrier . ${\displaystyle \{2,3,\dotsc ,12\}}$

The sum of uniformly distributed random variables that are not identical leads to a trapezoidal distribution , which in the discrete case comes about, for example, with the sum of the eyes of an ordinary cube and a regular tetrahedron that has the values ​​1, 2, 3 and 4 on its side surfaces : The sums 2 and 10 occur with the probability 1/24, the sums 3 and 9 with 2/24, 4 and 8 with 3/24, but 5, 6 and 7 with 4/24.

Individual proof

1. ↑ Ammar Grous: Analysis of Reliability and Quality Control: Fracture Mechanics 1. Iste Ltd. 2012, ISBN 978-1848214408