We have n balls with weights chosen from a gives set {w1, w2, ..., wc} with probabilities {Pr1, Pr2, ..., Prc} respectively. I am looking for a closed-form formula for the expected weight of total balls. As an example, I think for n=3 and c=2 the result should be like this:
'Expected Weight = 3*w1*(Pr1)^3+(2*w1+w2)*(Pr1)^2*Pr2+(w1+2*w2)*Pr1*(Pr2)^2+3*w2*(Pr2)^3'
Well if you only care about the weights expectation, you can use the additivity of the expectation. The expected weight of 1 ball is $w_1Pr_1 + w_2Pr_2+...+w_cPr_c$ therefore the expected weight of $n$ balls is that same number times $n$