Suppose that I know that a random variable $X$ has the distribution
$X \sim \cases{\text{Bernoulli}(p_1)\ \ \text{if}\ \ t > t_0 \\ \text{Bernoulli}(p_2)\ \ \text{if}\ \ 0 < t < t_0 }$
and that I collect pairs of data $(x_i,\ t_i)$. How should I go about when trying to make maximum likelihood estimations of $p_1, p_2$, and $t_0$ — and is this even possible?
Also, I'm actually more interested in the case where I already know what $p_1$ and $p_2$ are and looking for an estimation of $t_0$ (with a confidence interval, if possible), but the more general case above has piqued my interest as well.