 Solution

Multiply the Beta distribution density of $p_M$, the conditional binomial distribution mass of $X$, and the conditional Beta distribution density of $p_T$. With $B(a,b)$ the beta function, the full joint density, relative to counting measure in $x$ and Lebesgue measure in the two probabilities, isHere $x=0,\dots ,n$ and both probabilities lie in $(0,1)$. The printed joint expression omits the factor $1/B(\alpha +x,\beta +n-x)$. It is a constant when $x$ is fixed and only the two probabilities vary, but is not a constant for the full joint probability distribution. Keeping it is essential when summing over the latent variable.