Question #60816

Consider two subgroups $A$ and $B$ with sizes $N_A = 9 E[X_A]$ and $N_B = 5 E[X_B]$, where $X_A$ and $X_B$ are discrete random variables. The probability mass function (PMF) of $X_A$ is $P(X_A = k) = c_A \int_{k}^{k+2} \left[ \frac{x}{2} \right] dx$ for $k \in \{1, 3, 5\}$, and the PMF of $X_B$ is $P(X_B = k) = c_B \int_{0}^{k} \{x\} dx$ for $k \in \{1, 2, 3, 4\}$. Here, $[ \cdot ]$ and $\{ \cdot \}$ denote the greatest integer and fractional part functions, respectively. The mean of subgroup $A$ is $366$ and the mean of subgroup $B$ is $256$. Four quantities are defined: $\mu_{comb}$ is the combined mean of $A$ and $B$; $\sigma^2$ is the variance of $X_B$; $\Delta N = N_A - N_B$; and $E = 9 E[X_A]$. Which of the following values is INCORRECTLY matched with its quantity?