Let us work out some problems using exponential distributions. In reliability theory, it is common to assume that the lifespan of machines and components are random variables. That may suggest that the failure comes as Poisson, and the time between failures is exponentially distributed.
The time to failure of a tool follows an exponential distribution with the mean time between failures (MTBF) of 500 days. Calculate the probability this tool will fail before 500 days.
Since the mean time (for failure) is 500, lambda, the parameter is 1/500. Substituting for lambda and time, the probability becomes 1 – exp((1/500) *500) = 0.63 or 63%. The following R code also gives the same result.
pexp(500, 1/500)What is the probability that the tool will not fail for 1000 days?
The answer is 0.1353 or 13.53%
