# Estimating rewards for coin flip game, given the bias of the coin but not the outcome of the flip

I perform a series of $$N$$ coin flips, indexed $$i = 1, \ldots, N$$. I do not get to see the outcome of the coin flips, but for each one I know the probability of the coin being heads, $$p_i(H)$$. This changes for each coin flip, $$p_i(H) \neq p_j(H)$$ for $$i \neq j$$. Depending on the (hidden) outcome of the coin flip, I win an amount of money given by a random variable $$x$$ which is sampled from one of two fixed distributions, either $$q_H(x)$$ or $$q_T(x)$$, which do not vary with $$i$$. My task is to determine how much I expect to win if I get heads or tails. So for every $$i$$, I know the distribution of the coin that was flipped $$p_i(H)$$, and I know how much I win $$x_i$$. Can I esimate the expected value of $$x$$ given the coin was heads or tails, e.g. $$E[x \vert T] = \int\limits_x x \, q_T(x) dx$$?

A naive estimator of $$E[X \mid H]$$ might be the weighted average $$\dfrac{\sum p_i(H) x_i}{\sum p_i(H)}$$
and similarly of $$E[X \mid T]$$ might be $$\dfrac{\sum (1-p_i(H)) x_i}{\sum (1-p_i(H))}$$
• If the $$p_i(H)$$ are $$0$$ or $$1$$ (some of each), it will give you the obvious estimates for the conditional expectations
• If all the $$p_i(H)$$ are all equal (not $$0$$ nor $$1$$) it will give the same estimates for the two conditional expectations
• If the two conditional expectations are actually equal then the expected difference between the estimates of the conditional expectations will be $$0$$
• The difference between the estimates of two conditional expectations may be biased towards $$0$$ compared with the actual difference between the two conditional expectations