Basis Parameters

$X_i\left({\mu }_i,{\sigma }_i\right)\left(i=1,2,3,4\right)$

$\left({\mu }_i,{\sigma }_i\right)=\left(1,0.2\right),\left(4,0.5\right),\left(6,1.0\right),\left(10,1.5\right)$

Sample

(n = 1,000,000)

Theory

$Z=\Lambda \left({\mu }_Y,{\sigma }_Y^2\right)$

Basis Variables $X_i$

Sample Z

Sample Y

Sample Z(Y)

Y

Z

Normal

$\begin{array}{l}{m}_Z=240.01\\ s_Z=79.54\end{array}$

$\begin{array}{l}{m}_Y=5.43\\ s_Y=0.34\end{array}$

$\begin{array}{l}{M}_Z=240.48\\ S_Z=83.91\end{array}$

$\begin{array}{l}{\mu }_Y=5.48\\ {\sigma }_Y=0.33\end{array}$

$\begin{array}{l}{{\rm M}}_Z=253.05\\ {\Sigma }_Z=84.58\end{array}$

Uniform

$\begin{array}{l}{m}_Z=240.02\\ s_Z=79.53\end{array}$

$\begin{array}{l}{m}_Y=5.43\\ s_Y=0.33\end{array}$

$\begin{array}{l}{M}_Z=240.23\\ S_Z=82.11\end{array}$

—

—

Laplace

$\begin{array}{l}{m}_Z=240.01\\ s_Z=79.48\end{array}$

$\begin{array}{l}{m}_Y=5.42\\ s_Y=0.38\end{array}$

$\begin{array}{l}{M}_Z=243.01\\ S_Z=96.58\end{array}$

—

—

Log-Normal

$\begin{array}{l}{m}_Z=239.97\\ s_Z=79.50\end{array}$

$\begin{array}{l}{m}_Y=5.43\\ s_Y=0.32\end{array}$

$\begin{array}{l}{M}_Z=239.97\\ S_Z=79.49\end{array}$

$\begin{array}{l}{\mu }_Y=5.43\\ {\sigma }_Y=0.32\end{array}$

$\begin{array}{l}{{\rm M}}_Z=240.00\\ {\Sigma }_Z=79.60\end{array}$