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.

  • 1
    $\begingroup$ Thanks v much noob2, You are absolutely right. In fact, I now recall that the final equation, if rewritten, can be generalised to the kth moment of an rv taking on n values: $E[X^k]=\sum_x x^k p_X(x)$ $\endgroup$
    – markm
    Jun 14 '17 at 5:34

$$ E[X^2] = p[ln(1+f)]^2 + q[ln(1-f)]^2 $$

Is the standard way to compute the expectation of $X^2$, since in this case $X =\ln(1+Y_i f)$ has only two possible values: $\ln(1+f)$ with probability $p$ and $\ln(1-f)$ with probability q.

So it is quite a straightforward calculation.


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