La distribua funkcio de la standarda uniforma distribuo estas
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{\displaystyle F_{1}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\x,&{\text{se }}x\in \,]0,\,1]\\1,&{\text{se }}x>1{\text{.}}\\\end{cases}}}
Estu
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{\displaystyle F_{k}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\F_{k,\,1}(x),&{\text{se }}x\in \,]0,\,1]\\\cdots \\F_{k,\,j}(x),&{\text{se }}x\in \,]j-1,\,j]\\\cdots \\F_{k,\,k}(x),&{\text{se }}x\in \,]k-1,\,k]\\1,&{\text{se }}x>k\\\end{cases}}}
la distribu-funkcio de la sumo de k standard-uniform-distribuataj hazard-variabloj.
Do desegnas
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{\displaystyle F_{k,\,j}(x)}
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{\displaystyle ]j-1,\,j]}
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{\displaystyle Z_{k}\,}
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{\displaystyle F_{k}\,}
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{\displaystyle x\in \,]j-1,j]}
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Substitucio:
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{\displaystyle {\begin{aligned}F_{k+1,\,j}(x)&=P(\{Z_{k+1}<x\})\\&=P(\{Z_{k}+Z_{1}<x\})\\&=\int _{-\infty }^{\infty }P(\{Z_{k}<x-z_{1}\})\cdot P(\{Z_{1}\in \,[z_{1},z_{1}+dz_{1}[\})\\&=\int _{0}^{1}P(\{Z_{k}<x-z_{1}\})\cdot 1\,dz_{1}\\&=\int _{0}^{1}F_{k}(x-z_{1})\,dz_{1}&&{\text{Substitucio: }}y=x-z_{1}\\&=\int _{x-1}^{x}F_{k}(y)\,dy\\&=\int _{x-1}^{j-1}F_{k,\,j-1}(y)\,dy+\int _{j-1}^{x}F_{k,\,j}(y)\,dy.\end{aligned}}}
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{\displaystyle F_{k+1}\,}
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{\displaystyle F_{k}\,}
Ekzemple estas
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{\displaystyle F_{2}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\{\frac {x^{2}}{2}},&{\text{se }}x\in \,]0,\,1]\\{\frac {-x^{2}+4x-2}{2}},&{\text{se }}x\in \,]1,\,2]\\1,&{\text{se }}x>2{\text{.}}\\\end{cases}}}
La supre en formuloj kaj bildoj montrataj distribu-densoj
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{\displaystyle f_{k}\,}
![{\displaystyle F_{1}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\x,&{\text{se }}x\in \,]0,\,1]\\1,&{\text{se }}x>1{\text{.}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d9c062d92dd3f16f02dd49a65fa40b6165ec8d)
![{\displaystyle F_{k}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\F_{k,\,1}(x),&{\text{se }}x\in \,]0,\,1]\\\cdots \\F_{k,\,j}(x),&{\text{se }}x\in \,]j-1,\,j]\\\cdots \\F_{k,\,k}(x),&{\text{se }}x\in \,]k-1,\,k]\\1,&{\text{se }}x>k\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65f7b14dadfe024f7953dcbbff24e94734d1a8d5)
![{\displaystyle ]j-1,\,j]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e206abd4ae14e221e947347bc10fe00dd196bcce)
![{\displaystyle x\in \,]j-1,j]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/728aee4a8113098f98d8056df4a86161dbb476fe)
![{\displaystyle F_{2}(x)={\begin{cases}0,&{\text{se }}x\leq 0\\{\frac {x^{2}}{2}},&{\text{se }}x\in \,]0,\,1]\\{\frac {-x^{2}+4x-2}{2}},&{\text{se }}x\in \,]1,\,2]\\1,&{\text{se }}x>2{\text{.}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf63f7c81e9b6a0751e79c014f815cb3375ca363)
