There is a random variable and we know that it is either uniformly distributed on $(0, 1)$ or uniformly distributed on $(0, \frac{1}{2})$. Both cases are equally likely to be.
We are to guess the actual distribution. If we guess it correctly we are losing $0$ and if we guess it wrong we are losing $a: a > 0$.
We can check a value of the random variable. After one such check we are losing $c: c > 0$.
Now I am to prove that the optimal number of checkings should be $n$ such that it will minimize the expression $\frac{a}{2^{(n+1)}}+nc$.
Here is what I try to do. I start with noticing that I am to prove that total risk after $n$ experiments will be $\frac{a}{2^{(n+1)}}+nc$, because by proving that I will automatically prove the condition for the optimal number of checkings.
Also it is easy to notice that $nc$ term is the payment for $n$ experiments.
So, I am to prove that after $n$ experiments I will get the risk being equal $\frac{a}{2^{(n+1)}}$. Could you, please, help me to show that?
Thank you.