I am a research engineer. When I was analyzing my system, I came across the following problem. Since the problem involves dealing with random variables, I couldn't move so much other than running simulations (my probability knowledge is just moderate). But I still need some help to figure out an expression to make sense of my simulations and observations.
I have an N-element constant vector $\textbf{r} = [r_{1} \ r_{2}\cdot\cdot\cdot r_{N}]^{T}$ where $r_{i} \geq 0, i\in \{1,2,..,N\}$
I have an N-element random vector $\textbf{x} = [X_{1} \ X_{2}\cdot\cdot\cdot X_{N}]^{T}$ with each element $X_{i}$ being uniformly distributed over the interval $[a,b]$ where $a,b \geq 0$.
Also, I have the condition that the sum of all the elements in the random vector $\textbf{x}$ must be a constant $C$ such that
$$ C = \sum_{i=1}^{N}X_{i}$$
Now, I am trying to find the properties of the probability distribution resulting from the dot product of these two vectors.
$$ \textbf{x}^{T}\textbf{r} = X_{1}r_{1}+X_{2}r_{2}+\cdot\cdot\cdot+X_{N}r_{N}$$
Can I find any closed-form expression for the PDF, mean, and variance? I am particularly interested in the variance of this sum. I have run some computer simulations and generated random values using randfixedsum (link) and dot-product it with a constant vector $\textbf{r}$, and performed this experiment 10,000 times and investigated the histograms. I've made the following observations;
- $var(\textbf{x}^{T}\textbf{r})$ gets larger as $N$ decreases (for the same C).
- $var(\textbf{x}^{T}\textbf{r})$ gets smaller as $C$ decreases (for the same N).
- $var(\textbf{x}^{T}\textbf{r})$ gets smaller as $|a-b|$ decreases (for the same N and C).
I also have some visual observations that may not be so accurate but I'd like to list them to give you more intuition.
For a large difference between $a$ and $b$, the histogram of $\{X_{1} \ X_{2}\cdot\cdot\cdot X_{N}\}$ looks like an exponential distribution and the histogram of $\textbf{x}^{T}\textbf{r}$ looks like a gamma distribution.
For a small difference between $a$ and $b$, the histogram of $\{X_{1} \ X_{2}\cdot\cdot\cdot X_{N}\}$ looks like an uniform distribution and the histogram of $\textbf{x}^{T}\textbf{r}$ looks like a normal distribution.
I may have not used the correct notations for each expression since I am not a mathematician, but I tried to do my best to explain the problem. The problem itself could be simpler than I thought (maybe I miss something essential) or it could be very hard or even impossible to find a closed-form expression. I just need some analytical expression to support my observations above (if they are really true). I need some directions (paper, article, book, website, etc.) to look at for a further explanation for the problems of this sort. What can I say about the variance of this sum since its distribution looks very familiar?
I can provide more detail of my analysis and some example results if you need them. I can gladly appreciate your help.