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
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{\ displaystyle f (x) = {\ begin {cases} {\ frac {2 (xa)} {(ba) (ca)}}, & {\ text {if}} a \ leq x <c \\ { \ frac {2} {ba}}, & {\ text {if}} x = c \\ {\ frac {2 (bx)} {(ba) (bc)}}, & {\ text {if}} c <x \ leq b. \ end {cases}}}
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 .
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properties Distribution function
The distribution function
The distribution function is
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{\ displaystyle F (x) = {\ begin {cases} {\ frac {(xa) ^ {2}} {(ba) (ca)}}, & {\ text {if}} a \ leq x <c \\ {\ frac {ca} {ba}}, & {\ text {if}} x = c \\ 1 - {\ frac {(bx) ^ {2}} {(ba) (bc)}}, & {\ text {if}} c <x \ leq b. \ end {cases}}}
The inverse of the distribution function is
F.
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{\ 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
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E.
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{\ displaystyle \ operatorname {E} (X) = {\ frac {a + b + c} {3}}.}
For is the median given by
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{\ displaystyle m = b - {\ sqrt {(ba) (bc) / 2}}}
Pearson sense . Variance The variance of a triangular random variable results in
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Var
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{\ 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 .
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{\ displaystyle {\ sqrt {6}} (ba) / 12 \ approx 0 {,} 204 (ba)}
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{\ displaystyle (ba) / 6 \ approx 0 {,} 167 (ba)}
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{\ 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 .
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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.
literature Web links
Discrete univariate distributions
Continuous univariate distributions
Multivariate distributions
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![\ left [a, b \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30926fb280a9fdf66fd931e14d4363cb824feaa)
![x \ in \ left [a, b \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4dd3adabf0430c82fd37cb45576a1df8c60002)
