I am working through Thorpe's Ch 9 on the Kelly criterion.
On page 9 Thorpe states:
$$Var(ln(1+Y_if)] = p[ln(1+f)]^2 + q[ln(1-f)]^2 - m^2$$
Since $var(X) = E[X^2] - m^2$,
$$p[ln(1+f)]^2 + q[ln(1-f)]^2 = E[X^2]$$
Would it be correct to assume that $E[(ln\sum x_i))^k] = \sum p_i(ln(x_i))^k$ for rvs which can only take on two values? I am assuming this is the case but I am on a steep probability learning curve, so verification from someone with more experience would still be appreciated.