Basis Parameters

X i ( μ i , σ i ) ( i = 1 , 2 , 3 , 4 )

( μ i , σ i ) = ( 1 , 0.2 ) , ( 4 , 0.5 ) , ( 6 , 1.0 ) , ( 10 , 1.5 )

Sample

(n = 1,000,000)

Theory

Z = Λ ( μ Y , σ Y 2 )

Basis Variables X i

Sample Z

Sample Y

Sample Z(Y)

Y

Z

Normal

m Z = 240.01 s Z = 79.54

m Y = 5.43 s Y = 0.34

M Z = 240.48 S Z = 83.91

μ Y = 5.48 σ Y = 0.33

Μ Z = 253.05 Σ Z = 84.58

Uniform

m Z = 240.02 s Z = 79.53

m Y = 5.43 s Y = 0.33

M Z = 240.23 S Z = 82.11

Laplace

m Z = 240.01 s Z = 79.48

m Y = 5.42 s Y = 0.38

M Z = 243.01 S Z = 96.58

Log-Normal

m Z = 239.97 s Z = 79.50

m Y = 5.43 s Y = 0.32

M Z = 239.97 S Z = 79.49

μ Y = 5.43 σ Y = 0.32

Μ Z = 240.00 Σ Z = 79.60