Question

A product has an exponential time-to-failure distribution with a constant failure rate of 0.00006 per hour. The reliability of the product after 4000 hours of operation is

• A. 0.5866
• B. 0.6866
• C. 0.7866
• D. 0.8866

Hint:

Reliability for an exponentially distributed failure time decreases exponentially with time. The formula \( R(t) = e^{-\lambda t} \) can be used for such cases.

Answer 1

The Correct Option is C

The reliability \( R(t) \) of a product with an exponential time-to-failure distribution is given by the formula: \[ R(t) = e^{-\lambda t} \] where:
- \( \lambda \) is the failure rate,
- \( t \) is the time,
- \( R(t) \) is the reliability at time \( t \).
Here, \( \lambda = 0.00006 \, \text{per hour} \) and \( t = 4000 \, \text{hours} \). Substituting these values into the formula: \[ R(4000) = e^{-0.00006 \times 4000} = e^{-0.24}. \] Using a calculator: \[ e^{-0.24} \approx 0.7866. \] Thus, the reliability of the product after 4000 hours of operation is approximately 0.7866, so the correct answer is (C).