# Product of two Bernoulli random variables

I am having the following problem. I observe a random variable $$Y \sim \text{Bern}(p)$$, which is the product of two other independent random variables $$X_1 \sim \text{Bern}(p_1)$$ and $$X_2 \sim \text{Bern}(p_2)$$. I know that the original parameter is $$p = p_1 \cdot p_2$$. Is there any way to estimate $$p_1$$ and $$p_2$$?

Assuming that you are only observing the variable $$y$$, the probabilities $$p_1$$ and $$p_2$$ are not identifiable in your model; only the parameter $$p= p_1 \cdot p_2$$ is identifiable. This means that you essentially have no information with which to properly estimate $$p_1$$ and $$p_2$$.