I play a simple game which is modeled as follows:
I've a coin with heads probability $p$. I flip it and then give myself 1 coin with probability $p_w$ (if tails, I get nothing). Then I check the coin, if it's heads I stop, otherwise I continue until I get heads.
Let $R$ denote the number of coins I got when the game terminates. I'd like to calculate $E(R)$ and $Var(R)$. My attempt was as follows.
Let $N$ denote the number of rounds until I get a heads and $X$ denote the number of coins I get in a round. We have $R = N\cdot X$. We further have $N \sim Geom(p)$ and $X \sim Bern(p_w)$. Since $N$ and $X$ independent we can write,
$$ E(R) = E(N)\cdot E(X),$$ $$Var(R) = E(X)^2\cdot Var(N) + E(N)^2 \cdot Var(X) + Var(X)\cdot Var(N).$$
Finally, plugging the values yield. $$ E(R) = p_w / p $$ $$ Var(R) = \frac{p_w(2 - p - p_w)}{p^2} $$
I think the expected value is sound however, variance does't make any sense to me. I feel that the only thing that matters is the ratio $\frac{p_w}{p}$ but value of variance changes with actual values of probabilities.
In particular, we can have $E(R) - StdDev(R)$ negative in some cases which doesn't make any sense.
Am I overlooking something?
Yes, you are missing something. Your equation $R=NX$ is not coherent. You say $X$ is "the number of coins you get in a round" but what does that even mean? You have a random number $N$ different rounds and on each round you get some number $X_i$ of coins. Thus you should write $$ R= \sum_{i=1}^N X_i.$$
For help doing calculations where the number of elements in the sum is random, perhaps google Wald's theorem, which states that in fact your answer for expected value is correct. The secret is basically using conditional probability carefully.