Answer: 0.67

To find $E(X_1 | X_2 = 0.5)$, we need to compute the conditional expectation. First, we find the conditional probability density function $f(x_1 | x_2)$. **Step 1: Find the marginal density of $X_2$** $$f_{X_2}(x_2) = \int_{0}^{1} 8x_1x_2 \, dx_1 = 8x_2 \int_{0}^{1} x_1 \, dx_1 = 8x_2 \left[ \frac{x_1^2}{2} \right]_{0}^{1} = 8x_2 \cdot \frac{1}{2} = 4x_2$$ for $0 < x_2 < 1$. **Step 2: Find the conditional density $f(x_1 | x_2)$** $$f(x_1 | x_2) = \frac{f(x_1, x_2)}{f_{X_2}(x_2)} = \frac{8x_1x_2}{4x_2} = 2x_1$$ for $0 < x_1 < 1$ and $0 < x_2 < 1$. **Step 3: Compute the conditional expectation** $$E(X_1 | X_2 = x_2) = \int_{0}^{1} x_1 \cdot f(x_1 | x_2) \, dx_1 = \int_{0}^{1} x_1 \cdot 2x_1 \, dx_1 = \int_{0}^{1} 2x_1^2 \, dx_1$$ $$= 2 \left[ \frac{x_1^3}{3} \right]_{0}^{1} = 2 \cdot \frac{1}{3} = \frac{2}{3} \approx 0.6667$$ Therefore, $E(X_1 | X_2 = 0.5) = \frac{2}{3} \approx 0.67$, which corresponds to option D.

Author: Nikitesh Somanthe