Non-necessarily independent random variables $X\_1,~X\_2,~\cdots,~X\_n$ are supported on $[0,a\_1],~[0,a\_2],~\cdots,[0,a\_n]$ and with mean values $\mu\_1,~\cdots,~\mu\_n$ respectively, where all $a\_i$ and $\mu\_i$ are positive real numbers.


Assume that some system can only observe the sum of the above random variables, i.e., $S=\sum\_{i=1}^{n}X\_i$.


**How to decompose $S$ such that $S=\sum\_{i=1}^n V\_i$ and for any $i\in\{1,~2,~\cdots,n\}$:**


* $V\_i$ is supported on $[0,a\_i]$;
* The mean value $\mathbb{E}\left[V\_i \right]=\mu\_i$


An additional question is whether such the decomposition can be achieved by a series functions (i.e. $V\_i=\phi\_i(S))$.




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The above problem is edited as follows.


Assume that we know that $S$ is the sum of $n$ unknown nonnegative random variables $X\_1,~\cdots,~X\_n$ with known peak values ($a\_1,~\cdots,a\_n$) and mean values ($\mu\_1,~\cdots,~\mu\_n$) and the distribution of $S$ is available.


**How to decompose the random variable $S$ such that $V\_1,~\cdots,~V\_n$ satisfy the above-mentioned constraints?**



Decomposition of the sum of nonnegative random variables
(The OP's first clarification of the question asked about the case where the distribution of $S$ is not given.)


If you are not told the distribution of $S$, then it is not possible in general.


For example, suppose you are told $n=2$, $\mu\_1=\mu\_2=1/2$, $a\_1=1$, $a\_2=2$.


What will you do if you observe $S=1$?


It could be that $S=1$ always, say for example $(X\_1,X\_2)=(1/2,1/2)$ with probability $1$. Then on average your response to $S=1$ needs to divide up the mass equally between $V\_1$ and $V\_2$.


On the other hand it could be that
\begin{equation}
(X\_1,X\_2)=\begin{cases}
(1,0)& \text{with probability }1/4,\\
(1,2)& \text{with probability }1/4,\\
(0,0)& \text{with probability }1/2.
\end{cases}.
\end{equation}


So you'll observe $S=1$ a quarter of the time, and $S=3$ a quarter of the time, and when you observe $S=1$ you need to respond with $V\_1=1$, $V\_2=0$.


So without some information on, for example, the distribution of $S$, you do not know how to respond appropriately when you observe $S=1$.


