```{include} /../core/dataserver/attribute/introducing_tstochastic_attribute/compose_equation.md
```

The mean value of this newly generated law can be expressed, assuming that all pre-existing laws have a finite 
and defined expectation denoted $\lbrace \mu_j\rbrace_{1\leq j\leq N }$, as $\mu = \sum_{j=1}^{N}\frac{\omega_j \mu_j}{S}$  
where the sum of all weights is $S = \sum_{j=1}^{N} \omega_j$. As for the mean value, the variance of this newly generated 
law can be expressed, assuming that all pre-existing laws have a finite and defined expectation and variance, as done below in a very generic way.

```{math}
\begin{split}
  {\rm Var}_f & =  \mathbb{E}_f(x^2) - (\mathbb{E}_f(x))^2 \nonumber \\
              & =  \int x^2 f(x) dx - \bigg(\int xf(x)dx\bigg)^2, \;\, {\rm with} \;\, 
                f(x) = \sum_{j=1}^{N} \omega_j f_j(x) \;\, \rm{where} \;\, 
                \lbrace \omega_j\rbrace_{1\leq j \leq N} \in (\mathbb{R}^{+})^N \nonumber \\
              & =  \int \sum^{N}_{j=1} x^2 \frac{\omega_jf_j(x)}{S} dx - \bigg(\int \sum^{N}_{j=1} x 
                \frac{\omega_jf_j(x)}{S} dx\bigg)^2, \;\, \rm{where} \;\, S = \sum_{j=1}^N \omega_j \nonumber \\
              & = \frac{1}{S} \sum^{N}_{j=1} \omega_j \underbrace{\int x^2 f_j(x) dx}_{{\rm Var}_{f_j}(x) + 
                (\mathbb{E}_{f_j}(x))^2 } - \frac{1}{S^2}\bigg(\sum^{N}_{j=1} 
                \omega_j\underbrace{\int x f_j(x) dx}_{\mathbb{E}_{f_j}(x)}\bigg)^2\nonumber \\
              & = \frac{1}{S} \sum^{N}_{j=1} \omega_j (\sigma^2_j + \mu_j^2) - \frac{1}{S^2} \bigg(\sum^{N}_{j=1} 
                \delta_j\bigg)^2,\;\, {\rm where} \;\, \delta_j =  \omega_j \mu_j \;\, \forall j \in [1,N] \nonumber \\
              & =  \frac{1}{S} \sum^{N}_{j=1} \omega_j\sigma^2_j + \sum^{N}_{j=1} \frac{\delta_j^2}{S \omega_j} - 
                \frac{1}{S^2} \bigg[ \sum^{N}_{j=1} \delta^2_j + 2\sum_{1\leq i < j \leq N} \delta_i\delta_j  \bigg]\nonumber \\
              & =  \frac{1}{S} \sum^{N}_{j=1} \omega_j\sigma^2_j + \sum^{N}_{j=1} \frac{S-\omega_j}{S^2 \omega_j}  \delta_j^2 - 
                \frac{2}{S^2} \sum_{1\leq i < j \leq N} \delta_i\delta_j.
\end{split}
```

In the case of unweighted composition, this can be written as 
$\displaystyle{\rm Var}_f = \frac{1}{N}\sum^{N}_{j=1} \sigma^2_j + \frac{N-1}{N^2} \sum^{N}_{j=1} \mu_j^2 - 
\frac{2}{N^2} \sum_{1\leq i < j \leq N} \mu_i\mu_j$ .

```{include} /../core/dataserver/attribute/introducing_tstochastic_attribute/compose_figure.md
```