Many electronics follow a failure rate described by an exponential probability density function (PDF).

Question

Solar panels are advertised to last 20 years or longer, but panels made in China are failing at a higher rate. The time-to-failure of this device is usually exponentially distributed with mean 8 years. What is the probability of failure in the first 8 years?

Answer 1

Let X refer to the time-to-failure of a panel. We can write X~Exponential(8). As 8 is the mean, then we can say λ = 1/8. The PDF of this distribution is f(8, t) = (1/8)e^-(1/8)t, and integrating with respect to t gets the CDF, which is F(t) = -e^-(1/8)t. As we want to know the probability of failure in the first 8 years, evaluate P(X ≤ 8) = F(8) - F(0), which is approximately 0.6321.

The probability of failure in the first 8 years is 0.6321.