Answer: 0.36

**Step 1: Find the constant C** Since the probability density function must integrate to 1 over its entire domain: $$ \int_{-\infty}^{\infty} f(x)\,dx = 1 $$ $$ \int_{0}^{0.5} C e^{-x}\,dx = 1 $$ $$ C \left[-e^{-x}\right]_0^{0.5} = 1 $$ $$ C \left[-e^{-0.5} + e^{0}\right] = 1 $$ $$ C \left[-e^{-0.5} + 1\right] = 1 $$ $$ C (1 - e^{-0.5}) = 1 $$ $$ C = \frac{1}{1 - e^{-0.5}} \approx 2.5 $$ **Step 2: Find the 75th percentile** The 75th percentile $q_{75}$ satisfies: $$ P(X < q_{75}) = \int_0^{q_{75}} f_X(x)\,dx = 0.75 $$ $$ \int_0^{q_{75}} C e^{-x}\,dx = 0.75 $$ $$ C \left[-e^{-x}\right]_0^{q_{75}} = 0.75 $$ $$ C \left[-e^{-q_{75}} + 1\right] = 0.75 $$ $$ -e^{-q_{75}} + 1 = \frac{0.75}{C} $$ $$ e^{-q_{75}} = 1 - \frac{0.75}{C} $$ $$ -q_{75} = \ln\left(1 - \frac{0.75}{C}\right) $$ $$ q_{75} = -\ln\left(1 - \frac{0.75}{C}\right) $$ Substituting $C = \frac{1}{1 - e^{-0.5}}$: $$ q_{75} = -\ln\left(1 - 0.75(1 - e^{-0.5})\right) $$ $$ q_{75} = -\ln\left(1 - 0.75 + 0.75e^{-0.5}\right) $$ $$ q_{75} = -\ln\left(0.25 + 0.75e^{-0.5}\right) $$ $$ q_{75} \approx -\ln\left(0.25 + 0.75 \times 0.6065\right) $$ $$ q_{75} \approx -\ln\left(0.25 + 0.4549\right) $$ $$ q_{75} \approx -\ln(0.7049) $$ $$ q_{75} \approx 0.36 $$ Therefore, the 75th percentile is approximately 0.36.

Author: Nikitesh Somanthe