$ X_1 \sim Bin(n , p1) , X_2 \sim Bin(n , p2)$ are dependent R.V. and, p1 and p2 are such that:
$$ V \sim Uniform(a , b): $$ $$ p1 = P[V < r_1] $$ $$ p2 = P[V < r_2] $$
(1) Comment on the distribution of $X_2 | X_1$ , what simplifying assumptions would you make to make the calculations of the $E[X_2 | X_1]$ and $E[X_2 | X_1<s]$ tractable.
I know that if $X_1 \sim Bin(n , p)$ and $X_2|X_1 \sim Bin(n , q)$ then $X_2\sim Bin(n , pq)$.
So I am thinking that if we assume that $X_2 | X_1$ is Binomial and p2 < p1 and does that necessarily imply that $X_2 | X_1 \sim Bin(n, p2 / p1)$ in this case?