Question

A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean

μ=1000. Use this model to find the probability that a bulb fails within the first 200 hours.

A. 0.135
B. 0.183
C. 0.231
D. 0.368

Answer 1

To find the probability that a bulb fails within the first 200 hours, we can use the exponential density function with mean μ=1000. The probability is approximately 0.18127.

Step-by-step explanation:

To find the probability that a bulb fails within the first 200 hours, we can use the exponential density function with mean μ=1000. The probability density function (pdf) for an exponential distribution with mean μ is given by f(x) = (1/μ)e^(-x/μ). In this case, μ=1000, so f(x) = (1/1000)e^(-x/1000).

To find the probability that a bulb fails within the first 200 hours, we need to integrate the pdf from 0 to 200. This will give us the cumulative distribution function (cdf) for the exponential distribution.

∫[0 to 200] (1/1000)e^(-x/1000) dx = [-e^(-x/1000)] [0 to 200] = -e^(-200/1000) - (-e^(-0/1000)) = 1 - e^(-0.2) ≈ 0.18127.

Therefore, the probability that a bulb fails within the first 200 hours is approximately 0.18127.