Two distinct coins, $A$ and $B$, are tossed three times. $A$ is fair coin, that is, $P(\textrm{heads}) = P(\textrm{tails}) = 1/2$. $B$ is not a fair coin, with, $P(\textrm{heads}) = 1/4$ and $P(\textrm{tails}) = 3/4$. Let $X$ be the $r.v.$ that denotes the number of heads from $A$, and $Y$ be the $r.v.$ that denotes the number of heads from $B$.
Determine the join pmf of $X$ and $Y$.
My reasoning:
Clearly these are two random variables with binomial distributions.Let $Z = X + Y$. Then $Z \sim \textrm{Binom}(n = 3, p= \frac{0.5 + 0.25}{2} = 0.375)$
Your answer is wrong. The question is asking for the joint pmf, which, since $X,Y$ are independent, is just a product of the individual pmf's.
To find such a thing, you should be looking at $Z=(X,Y)$, not $Z=X+Y$, which is a convolution...