Review1 of Liber De Ludo Aleae (Book on Games of Chance) by ...

To illustrate this transformation, consider the two variable case, using:
Then
t + t = λ + λ + λ + 2λλ
+ λ
2 2 2 2 2 2
1 2 1 2 1 1 2 2
⎛ 2 2 2 2 ⎞
⎛ t1
⎞ ⎜ 2
L ⎟⎛
λ1
⎞
⎜ ⎟ ⎜ 2 2 2 2 ⎟⎜
⎟
⎜ ⎟ ⎜ 3 3 1 3 1 3 1 ⎟⎜
⎟
⎜ t ⎟
2 ⎜ 0 ⋅ ⋅ L ⋅ ⎟⎜
λ ⎟
2
⎜ ⎟ ⎜ 2 2 3 2 3 2 3 ⎟⎜
⎟
⎜ ⎟ ⎜ 4 4 1 4 1 ⎟⎜
⎟
⎜ t ⎟
3 = ⎜ 0 0
⋅ L ⋅ ⎟⎜
λ ⎟
3
⎜ ⎟ ⎜ 3 3 4 3 4 ⎟⎜
⎟
⎜ ⎟ ⎜ ⎟⎜
⎟
⎜ ⎟ ⎜ ⎟⎜
⎟
⎜
M
⎟ ⎜
M
⎜ O
⎟ ⎟
⎜ ⎟ ⎜ ⎟⎜
⎟
⎜ ⎟ n 1
t ⎜ ⎜ ⎟
n−1 0 0 0 0
⎟
⎝ ⎠ ⎜ L
⋅ ⎟⎝λn−1⎠
⎝ n−1n⎠ t
2
1 2( λ1
2 )
t
= +
=
3 λ
2
2 2
2 2 2
1 2 3
36
λ
= λ + λ + λ , since λ3 =−λ1− λ2.
The determinant <strong>of</strong> the transformation is √n, noting that the associated matrix is upper triangular, with
product <strong>of</strong> the diagonal terms:
2 ⋅
3
⋅
2
4
⋅L⋅ 3
n
n −1
= 2⋅ 1
⋅
2
3⋅ 1
⋅L⋅ 3
n−1⋅ 1
⋅
n −1
n
= n
Then the probability that [tt] is between u and u + du is given by
⎡ h ⎤
⎢ ⎥
⎣ π ⎦
n−1
∫ ∫
2
−h
[ tt]
2
L e dt Ldt
1 n−1
Helmert then refers to a result he obtained in 1875: The probability that the sum [tt] <strong>of</strong> n-1 true errors
equals u, is given by:
Γ
h
n−1
n − 1
( )
2
n−3
2
2
−hu
⋅u ⋅e
du