Answer: 39.35%

## Explanation For a Poisson process with an average of 6 defaults per year, we need to find the probability that the next default occurs within one month. **Step-by-step calculation:** 1. **Convert annual rate to monthly rate**: - Annual default rate λ = 6 defaults/year - Monthly default rate = 6/12 = 0.5 defaults/month 2. **Time between defaults in Poisson process**: - In a Poisson process, the time between events follows an exponential distribution - The probability that the time to next default is less than or equal to t is: \[ P(T \leq t) = 1 - e^{-\lambda t} \] 3. **Apply to our case**: - λ = 0.5 defaults/month - t = 1 month - \[ P(T \leq 1) = 1 - e^{-0.5 \times 1} = 1 - e^{-0.5} \] 4. **Calculation**: \[ 1 - e^{-0.5} = 1 - 0.6065 = 0.3935 = 39.35\% \] **Verification**: - The exponential distribution parameter is λ = 0.5 - The probability density function is f(t) = 0.5e^{-0.5t} - Integrating from 0 to 1 gives the cumulative probability Therefore, the correct answer is **C. 39.35%**.

Author: LeetQuiz .