Basis Parameters ${X}_{i}\left({\mu }_{i},{\sigma }_{i}\right)\text{\hspace{0.17em}}\left(i=1,2,3,4\right)$ $\left({\mu }_{i},{\sigma }_{i}\right)=\left(1,0.2\right),\left(4,0.5\right),\left(6,1.0\right),\left(10,1.5\right)$ 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 $\begin{array}{l}{m}_{Z}=240.01\\ {s}_{Z}=79.54\end{array}$ $\begin{array}{l}{m}_{Y}=5.43\\ {s}_{Y}=0.34\end{array}$ $\begin{array}{l}{M}_{Z}=240.48\\ {S}_{Z}=83.91\end{array}$ $\begin{array}{l}{\mu }_{Y}=5.48\\ {\sigma }_{Y}=0.33\end{array}$ $\begin{array}{l}{Μ}_{Z}=253.05\\ {\Sigma }_{Z}=84.58\end{array}$ 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 $\begin{array}{l}{m}_{Z}=239.97\\ {s}_{Z}=79.50\end{array}$ $\begin{array}{l}{m}_{Y}=5.43\\ {s}_{Y}=0.32\end{array}$ $\begin{array}{l}{M}_{Z}=239.97\\ {S}_{Z}=79.49\end{array}$ $\begin{array}{l}{\mu }_{Y}=5.43\\ {\sigma }_{Y}=0.32\end{array}$ $\begin{array}{l}{Μ}_{Z}=240.00\\ {\Sigma }_{Z}=79.60\end{array}$