Question

Suppose that the MTBF for a new automated manufacturing system follows an exponential distribution with a mean of 12.8 hours. What is the probability that the automated manufacturing system runs for more than 18 hours without a failure?

Answer 1

The probability that the automated manufacturing system runs for more than 18 hours without a failure is 0.547.

Step-by-step explanation:

To find the probability that the automated manufacturing system runs for more than 18 hours without a failure, we need to use the exponential distribution. The mean time between failures (MTBF) for the system is given as 12.8 hours. The exponential distribution has a probability density function (PDF) of f(x) = (1/mean) * e^(-x/mean), where x is the time without failure. Since we are interested in the probability of the system running for more than 18 hours without failure, we need to calculate the integral of the PDF from 18 to infinity.

First, we calculate the value of lambda, the rate parameter of the exponential distribution, which is equal to 1/mean. In this case, lambda = 1/12.8 = 0.078. The integral of the PDF from 18 to infinity can be calculated as the complement of the cumulative distribution function (CDF) from 0 to 18. Using a calculator or statistical software, we can find that the CDF from 0 to 18 is 0.453. Therefore, the probability that the automated manufacturing system runs for more than 18 hours without a failure is 1 - 0.453 = 0.547.