Expected value of the product of the sum of a specific distribution

Question

How can we find the value of the following term,

$$ E[\prod_{i = 1}^{L}{\sum_{j = 1}^{K}{a_{ij}}}] $$

i.e., the expected value of the product of the sum of $a_{ij}$'s where $a_{ij}$ is a random variable drawn from a probability distribution $f(x)$. How can I compute the value for a general $f(.)$? What if $f(x) = \frac{1}{\sqrt{x}}$ and $c_1 \le x \le c_2$?

Answer 1

If the $a_{ij}$ are not only identically distributed but also independent, your expectaton is $(K\alpha)^L$ where $\alpha=E(a_{ij})$.

Since the independence assumption is only needed to disentangle the sums $b_i=\displaystyle\sum_{j=1}^Ka_{ij}$ but not to compute $E(b_i)=K\alpha$, this assumption can be relaxed to the $b_i$s being $L$ independent random variables.

Answer 2

Since $E(XY) = E(X)E(Y)$ for random and independent variables as can be seen by: $$\int_x\int_y\;xy\;f(x)g(y)\;dx\;dy = \int_x xf(x)\;dx\int_y yf(y)\;dy$$ Didier Pau's answer is correct: $(K\;E(a))^L$