Given a set $A$ of $n=kl$ positive real numbers, let $a_{ij}$ be a random number removed from the set $A$ (without replacement) for $i=1 \dots k$ and $j=1 \dots l$. What is the distribution and expected value of the following expression?
$$ \prod_{i=1}^{k} \sum_{j=1}^{l} a_{ij} $$
Note: Using a set of 30 numbers for $l=3$ and $k=10$, I generated 10000 samples and computed the above formula for it. The following figure shows the distribution of the samples.
My question is that: how can I find the above distribution and expected value analytically?
Update
To makes the things clear, I past the code I used to generate the histogram here. This is a Mathematica code. The code generates a set of 30 random numbers between 0 and 1 (variable "nums"). Then it does the following 10000 times to generate the numbers for the smooth histogram: Shuffle the numbers in "nums" and assign it to "rands"; split the numbers in "rand" into 10 groups of 3 numbers; compute the sum of each 3 numbers; multiply the 10 summations to have the result.
SmoothHistogram@Module[{nums = RandomReal[1, {3*10}], rands},
Table[
rands = RandomSample@nums;
Times @@ Total /@ Table[rands[[{i, i + 1, i + 2}]], {i, 1, Length[nums], 3}],
{10000}
]
]
R
code in my reply (to simulate a setx
of its values): one draws $l$ groups of $k$ values from a set $A$, without replacement, and forms the product of the sums of those values. $\endgroup$