# Triangular distribution

The triangular distribution (or Simpson distribution , after Thomas Simpson ) is a continuous probability distribution that is used in probability theory and statistics .

## definition

The triangular distribution is defined by the probability density function defined on the interval${\ displaystyle \ left [a, b \ right]}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {2 (xa)} {(ba) (ca)}}, & {\ text {if}} a \ leq x The parameters (minimum value), (maximum value) and (most likely value) determine the shape of the triangular distribution ( and ). The graph of the density function looks like a triangle and gives this distribution its name. The axis shows the density of the respective probability for a value . ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle a ${\ displaystyle a \ leq c \ leq b}$ ${\ displaystyle y}$ ${\ displaystyle x \ in \ left [a, b \ right]}$ ## properties

### Distribution function

${\ displaystyle F (x) = {\ begin {cases} {\ frac {(xa) ^ {2}} {(ba) (ca)}}, & {\ text {if}} a \ leq x The inverse of the distribution function is

${\ displaystyle F ^ {- 1} (y) = {\ begin {cases} a + {\ sqrt {y (ba) (ca)}}, & {\ text {if}} 0 \ leq y \ leq {\ frac {(ca)} {(ba)}} \\ b - {\ sqrt {(ba) (bc)}} {\ sqrt {(1-y)}}, & {\ text {if}} {\ frac {(ca)} {(ba)}} \ leq y \ leq 1 \ end {cases}}}$ ### Expected value and median

The expected value of a triangular random variable is ${\ displaystyle X}$ ${\ displaystyle \ operatorname {E} (X) = {\ frac {a + b + c} {3}}.}$ For is the median given by ${\ displaystyle bc> ca}$ ${\ displaystyle m}$ ${\ displaystyle m = b - {\ sqrt {(ba) (bc) / 2}}}$ . In this case the median is smaller than the expected value; d. H. the distribution is skewed to the right in the Pearson sense .

### Variance

The variance of a triangular random variable results in ${\ displaystyle X}$ ${\ displaystyle \ operatorname {Var} (X) = {\ frac {a ^ {2} + b ^ {2} + c ^ {2} -ab-ac-bc} {18}} = {\ frac {( ab) ^ {2} + (bc) ^ {2} + (ac) ^ {2}} {36}}.}$ ## Relationship to other distributions

### Sum of uniformly distributed random variables

The sum of two identical, independent and continuously uniformly distributed random variables is triangularly distributed with , standard deviation , mean absolute deviation and interquartile range . ${\ displaystyle bc = ca}$ ${\ displaystyle {\ sqrt {6}} (ba) / 12 \ approx 0 {,} 204 (ba)}$ ${\ displaystyle (ba) / 6 \ approx 0 {,} 167 (ba)}$ ${\ displaystyle (1 - {\ sqrt {2}} / 2) (ba) \ approx 0 {,} 293 (ba)}$ ### Amount of the difference between uniformly distributed random variables

The amount of the difference between two identical, independent and continuously equally distributed random variables is triangularly distributed with . ${\ displaystyle | X_ {1} -X_ {2} |}$ ${\ displaystyle a = c = 0}$ ### Trapezoidal distribution

The triangular distribution is a special case of the trapezoidal distribution .

### Discrete triangular distribution

The continuous triangular distribution can be understood as the limit value of a discrete triangular distribution.