Answer: 31.250%

## Explanation To find the probability that the bond price is between $8.00 and $9.00, we need to integrate the probability density function (pdf) over the interval [8, 9]. Given: $$f(x) = \frac{x}{8} - 0.75 \quad \text{for } 6 \leq x \leq 10$$ **Step 1: Set up the integral** $$P(8 \leq x \leq 9) = \int_{8}^{9} \left(\frac{x}{8} - 0.75\right) dx$$ **Step 2: Evaluate the integral** $$\int \left(\frac{x}{8} - 0.75\right) dx = \frac{x^2}{16} - 0.75x + C$$ **Step 3: Apply the limits** $$\left[\frac{x^2}{16} - 0.75x\right]_{8}^{9} = \left(\frac{9^2}{16} - 0.75 \times 9\right) - \left(\frac{8^2}{16} - 0.75 \times 8\right)$$ **Step 4: Calculate** At x = 9: $$\frac{81}{16} - 6.75 = 5.0625 - 6.75 = -1.6875$$ At x = 8: $$\frac{64}{16} - 6 = 4 - 6 = -2$$ **Step 5: Find the difference** $$(-1.6875) - (-2) = -1.6875 + 2 = 0.3125$$ **Step 6: Convert to percentage** $$0.3125 \times 100\% = 31.25\%$$ Therefore, the probability that the bond price is between $8.00 and $9.00 is **31.250%**, which corresponds to option C. **Verification**: We can verify this is a valid probability by checking that the pdf integrates to 1 over the entire domain [6,10]: $$\int_{6}^{10} \left(\frac{x}{8} - 0.75\right) dx = \left[\frac{x^2}{16} - 0.75x\right]_{6}^{10} = \left(\frac{100}{16} - 7.5\right) - \left(\frac{36}{16} - 4.5\right) = (6.25 - 7.5) - (2.25 - 4.5) = (-1.25) - (-2.25) = 1$$

Author: LeetQuiz .