Question

**Given:** $X = \{1, 2, 3, \ldots, 19\}$, $Y = \{ax + b : x \in X\}$, $\text{Mean}(Y) = 30$, $\text{Var}(Y) = 750$. **Step 1: Compute Mean and Variance of $X$.** $$\text{Mean}(X) = \frac{\displaystyle\sum_{x=1}^{19} x}{19} = \frac{\frac{19 \times 20}{2}}{19} = 10$$ $$\sum_{x=1}^{19} x^2 = \frac{19 \times 20 \times 39}{6} = 2470$$ $$\text{Var}(X) = \frac{\sum x^2}{19} - \left(\text{Mean}(X)\right)^2 = \frac{2470}{19} - 100 = 130 - 100 = 30$$ **Step 2: Apply linear transformation properties.** For $Y = aX + b$: $$\text{Mean}(Y) = a \cdot \text{Mean}(X) + b \implies 10a + b = 30 \quad \cdots (1)$$ $$\text{Var}(Y) = a^2 \cdot \text{Var}(X) \implies 30a^2 = 750 \implies a^2 = 25 \implies a = \pm 5$$ **Step 3: Find the corresponding values of $b$.** - If $a = 5$: from $(1)$, $\;50 + b = 30 \implies b = -20$ - If $a = -5$: from $(1)$, $\;-50 + b = 30 \implies b = 80$ **Step 4: Sum of all possible values of $b$.** $$b_1 + b_2 = -20 + 80 = 60$$ **Answer: (D) $60$**