The life of a certain type of device has an advertised failure of rate .01 per hour. The failure rate is constant and the exponential distribution applies.
(a)What is the mean time to failure?
(b)What is the probability that 150 hours will pass before a failure is observed?
(c)What is the probability that 3 of the next 6 failures observed will occur after 150 hours have passed?
(d)What is the probability that at most 3 failures will occur in 150 hours?
To solve these questions, we will use the exponential distribution, which is commonly used to model the time until a specific event occurs. The exponential distribution has a constant failure rate, and its mean is equal to the inverse of the failure rate. Let's solve each part step by step:
(a) The mean time to failure can be calculated by taking the inverse of the failure rate. In this case, the failure rate is 0.01 per hour. So, the mean time to failure (denoted as mu) is given by:
mu = 1 / failure rate = 1 / 0.01 = 100 hours
Therefore, the mean time to failure is 100 hours.
(b) To find the probability that 150 hours will pass before a failure is observed, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of the exponential distribution is given by:
CDF(t) = 1 - e^(-lambda * t)
Where lambda is the failure rate, and t is the time at which we want to calculate the probability.
In this case, we want to find the probability that 150 hours will pass before a failure occurs. So, we substitute t = 150 into the CDF equation:
CDF(150) = 1 - e^(-0.01 * 150)
Using a calculator, we can evaluate this expression to find the probability.
(c) To find the probability that 3 of the next 6 failures observed will occur after 150 hours have passed, we can use the probability mass function (PMF) of the exponential distribution. The PMF of the exponential distribution is given by:
PMF(t) = lambda * e^(-lambda * t)
We want to calculate the probability that out of the next 6 failures, exactly 3 will occur after 150 hours. This can be calculated using the binomial distribution, but we need the probability of one failure occurring after 150 hours.
To find that probability, we can subtract the cumulative distribution function (CDF) value at 150 hours from 1. Let p be the probability of a failure occurring after 150 hours, then p = 1 - CDF(150).
Next, we can use the binomial distribution formula with n = 6 (number of trials), p (probability) calculated in the previous step, and k = 3 (number of successes):
Probability of 3 failures occurring after 150 hours = binomial(n, k) * p^k * (1-p)^(n-k)
(d) To find the probability that at most 3 failures will occur in 150 hours, we need to calculate the probability of 0, 1, 2, and 3 failures separately, and then sum them up. We can again use the exponential distribution's PMF and the binomial distribution formula to calculate these probabilities.