Answer: $6,931

The median return is calculated by solving for m in the equation: $$ P(X < m) = \int_{-\infty}^{m} f_X(x)\,dx = F(m) = 0.5 $$ For the given exponential distribution: $$ \int_{0}^{m} 0.0001e^{-0.0001x}\,dx = 0.5 $$ Solving the integral: $$ \left[ -e^{-0.0001x} \right]_0^m = 0.5 $$ $$ -e^{-0.0001m} + e^{0} = 0.5 $$ $$ -e^{-0.0001m} + 1 = 0.5 $$ $$ -e^{-0.0001m} = -0.5 $$ $$ e^{-0.0001m} = 0.5 $$ Taking natural logarithm: $$ -0.0001m = \ln(0.5) $$ $$ m = \frac{\ln 0.5}{-0.0001} = 6931.47 $$ Therefore, the median return is approximately $6,931, which corresponds to option A.

Author: Nikitesh Somanthe