average and standard deviation of distribution given by sum with exponential form

Question

We have a distribution with D sample given by:

$F(k)=\sum_{j=1}^N w_j(k)\times p_j(k)$

so k=1...D generating samples F(1)...F(D)

and

$\sum_{j=1}^N w_j(k) = 1$

also, $\ w$ has the form

$w \equiv A^B$

so

$F(k)=\sum_{j=1}^N \ [A_j(k)]^B\times p_j(k)$

Question: What is the average and standard deviation of distribution F given as a function of B.

Approximations are welcome and if the ratio $\ Average(F) \over Standard Deviation(F)$ offers a simplifications would be great.