For a usual coin with probability of heads $P(H)$, the distribution of the numbers of heads after $n$ tosses follows a binomial distribution. The coin can be biased or unbiased.
But suppose there's another variable, $\beta$, that controls whether heads outcomes are allowed to begin with. $\beta$ ranges from 0 to 1. $\beta = 0.6$ means 60% of the time, the outcome is forced to be not heads (i.e. tails) regardless of the natural outcome of the coin toss. $\beta = 0$ essentially means this factor is removed, coin can land either way naturally. $\beta = 1$ means 100% of the time, outcome is always tails.
What distribution would the numbers of heads after $n$ tosses follow, with respect to both the natural probability $P(H)$ and the control factor $\beta$?
Initially I thought it could be bivariate normal distribution, but that requires 2 independent variables, and I'm not sure $P(H)$ is independent of $\beta$ in this case.