Question #40984

Let $X$ and $Y$ be two independent discrete random variables. The probability mass function of $X$ is $P(X=k) = \alpha \left[ \frac{k^3}{3} \right]$ for $k \in \{1, 2, 3, 4\}$, and the probability mass function of $Y$ is $P(Y=k) = \beta \left( k^2 + \left[ \frac{k}{2} \right] \right)$ for $k \in \{1, 2, 3, 4, 5\}$, where $[ \cdot ]$ denotes the greatest integer function. Here, $\alpha$ and $\beta$ are constants such that the sum of probabilities for each distribution is $1$. Two subgroups $A$ and $B$ have a number of elements given by $N_A = 256 E[X]$ and $N_B = 366 E[Y]$, where $E[\cdot]$ denotes the expected value. If the mean of subgroup $A$ is $321$ and the mean of subgroup $B$ is $20$, what is the combined mean of the two subgroups?