Suppose r successes were obtained in a sequence of m Bernoulli trials with success probability $\displaystyle {\Theta}_1$, and that in an independent second sequence of n Bernoulli trials with success probability $\displaystyle {\Theta}_2$, there were s successes.

Show that the maximum of the log-likelihood of all obervations is

$\displaystyle In{\binom{m}{r}} + In{\binom{n}{s}} +

rIn(r) + sIn(s) + (m-r)In(m-r) +$$\displaystyle (n-s)In(n-s) - mIn(m) - nIn(n)$.

Is there a formula for this and its then just a case of plugging in the variables?