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An investment account has a return X, up to a maximum profit of $10,000. The probability density function for X is: $ f(x) = \begin{cases} 0.0001e^{-0.0001x}, & x \geq 0 \\ 0, & \text{elsewhere} \end{cases} $ Calculate the median return. | Financial Risk Manager Part 1 Quiz - LeetQuiz

Explanation:

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.

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An investment account has a return X, up to a maximum profit of `$10`,000. The probability density function for X is:
$f(x) = \begin{cases} 0.0001e^{-0.0001x}, & x \geq 0 \\ 0, & \text{elsewhere} \end{cases}$
Calculate the median return.

Comments (0)