สำหรับ i = 1 , 2 , … , n {\displaystyle i=1,2,\ldots ,n\,} ให้ X i = | a i − i | {\displaystyle X_{i}=|a_{i}-i|\,} จาก law of total expectation เราได้ว่า
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ให้ 
จาก law of total expectation เราได้ว่า
![{\displaystyle E[X_{i}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c351ad22dc33eed67b108cbc3330c9e837f50e73)
![{\displaystyle \Pr(a_{i}<i)E[X_{i}\ |\ a_{i}<i]+\Pr(a_{i}=i)E[X_{i}\ |\ a_{i}=i]+\Pr(a_{i}>i)E[X_{i}\ |\ a_{i}>i]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/406236064de1e0fa6d93713b554e307ab2a74d99)
![{\displaystyle \Pr(a_{i}<i)E[i-a_{i}\ |\ a_{i}<i]+\Pr(a_{i}=i)\cdot 0+\Pr(a_{i}>i)E[a_{i}-i\ |\ a_{i}>i]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3abfe26561cb6a4df99be05f535157234c30a2eb)
![{\displaystyle {\frac {i-1}{n}}E[i-a_{i}\ |\ a_{i}<i]+{\frac {n-i}{n}}E[a_{i}-i\ |\ a_{i}>i]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf2101084720c5fc7f2f52849cac1435423f2d0)
![{\displaystyle {\frac {i-1}{n}}{\big (}i-E[a_{i}\ |\ a_{i}<i]{\big )}+{\frac {n-i}{n}}{\big (}E[a_{i}\ |\ a_{i}>i]-i{\big )}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0b6d1d9ac0789026d67379165f82eded770536)
![{\displaystyle {\frac {1}{2n}}[(i-1)i+(n-i)(n-i+1)]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ccf46e0bdaba00889c3b325cbd0c6800b7e7e8)
![{\displaystyle E{\bigg [}\sum _{i=1}^{n}|a_{i}-i|{\bigg ]}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32ab0fa9ea320cd8d2ed2682307a351cc8e0f692)
![{\displaystyle \sum _{i=1}^{n}E[X_{i}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e78038c0c80a4e8733ff94d3fe24e9198cdb6a0)
![{\displaystyle \sum _{i=1}^{n}{\frac {1}{2n}}[(i-1)i+(n-i)(n-i+1)]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc244c41ed3fb1458275a3fc588b99f0e8b6b26)
