International Scholarly Research Notices

Research Article

On the Order Statistics of Standard Normal-Based Power Method Distributions

Table 9

Expected values of order statistics for 𝑝 1 ( 𝑍 ) = 𝑍 for 𝑛 = 8 .
𝐸 [ 𝑝 1 ( 𝑍 ) 5 ∶ 8 ] = − 2 1 0 𝜀 2 𝜋 5 / 2 + 9 8 0 𝜀 3 𝜋 7 / 2 − 2 1 0 t a n − 1 √ ( 1 / 2 ) 2 𝜋 5 / 2 + 9 8 0 t a n − 1 √ ( 1 / 2 ) 3 𝜋 7 / 2 = 0 . 1 5 2 5 1 4 3 9 …
𝐸 [ 𝑝 1 ( 𝑍 ) 6 ∶ 8 ] = 5 4 6 𝜀 2 𝜋 5 / 2 − 5 8 8 𝜀 3 𝜋 7 / 2 − 8 4 t a n − 1 √ ( 1 / 2 ) 𝜋 3 / 2 + 5 4 6 t a n − 1 √ ( 1 / 2 ) 2 𝜋 5 / 2 − 5 8 8 t a n − 1 √ ( 1 / 2 ) 3 𝜋 7 / 2 = 0 . 4 7 2 8 2 2 4 9 …
𝐸 [ 𝑝 1 ( 𝑍 ) 7 ∶ 8 ] = − 1 4 √ 𝜋 − 4 6 2 𝜀 2 𝜋 5 / 2 + 1 9 6 𝜀 3 𝜋 7 / 2 + 1 6 8 t a n − 1 √ ( 1 / 2 ) 𝜋 3 / 2 − 4 6 2 t a n − 1 √ ( 1 / 2 ) 2 𝜋 5 / 2 + 1 9 6 t a n − 1 √ ( 1 / 2 ) 3 𝜋 7 / 2 = 0 . 8 5 2 2 2 4 8 6 …
𝐸 [ 𝑝 1 ( 𝑍 ) 8 ∶ 8 ] = 1 4 √ 𝜋 + 1 2 6 𝜀 2 𝜋 5 / 2 − 2 8 𝜀 3 𝜋 7 / 2 − 8 4 t a n − 1 √ ( 1 / 2 ) 𝜋 3 / 2 + 1 2 6 t a n − 1 √ ( 1 / 2 ) 2 𝜋 5 / 2 − 2 8 t a n − 1 √ ( 1 / 2 ) 3 𝜋 7 / 2 = 1 . 4 2 3 6 0 0 3 0 …