Can someone help understand a function?
I saw this video about The "Just One More" Paradox and there is this one function which the creator showed and then simplified it.
I've been trying to get to the simplified version of the function myself but couldn't and got stuck midway.
My final answer is: $f(q + p)= \frac{p}{a} -\frac{q}{b} $
The function I was trying to get to: $f^* = \frac{p}{a} -\frac{q}{b} $
The creator also used an * above the $f$ in the function, does it have a meaning?
If someone can show the way of how he got to the simplified function, it'd be great
The link is timestamped with the formula https://youtu.be/_FuuYSM7yOo?t=463
Mkay
1 Answer
We want to find $f^*$ that maximizes $r$. At the maximum point, the derivative of $r$ with respect to $f$ has to be zero.
\[r=(1+fb)^p(1-fa)^q.\]
Then
\[\ln r=p\ln(1+fb)+q \ln (1-fa).\]
Then
\[\frac{r'}{r}=p \frac{b}{1+fb}-q\frac{a}{1-fa}=\frac{pb(1-fa)-aq(1+fb)}{(1+fb)(1-fa)}=0.\]
Hence
\[pb(1-fa)-aq(1+fb)=0\]
\[\Rightarrow pb-fabp-aq-abqf=0\]
\[\Rightarrow fab(p+q)=pb-aq\]
\[\Rightarrow f^*=f (a+b)=\frac{pb-aq}{ab}=\frac{p}{a}-\frac{q}{b}.\]
So you have the correct final answer.
Here $f^*$ is the fraction of the assets to apply to the security. It is the fraction of your wealth you would like to invest. See Kelly Criterion:
https://en.wikipedia.org/wiki/Kelly_criterion
Erdos
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f^* does not have a meaning. It is simply a function denoted by f^*. Your second question about the simplified function is a bit time-consuming and should come with a fair bounty.