the time between unplanned shutdowns of a power plant has an exponential distribution with a mean of 20 days. Find the probability that the time between two unplanned shutdow is a) less than 14 days B) more than 21 days c) less than 7 days

please answer the questions I posted my homework is due by thursday 1/13/11

To find the probabilities in this scenario, we can use the exponential distribution formula. The exponential distribution is often used to model the time between events in a Poisson process, such as the time between unplanned shutdowns of a power plant.

The probability density function (PDF) of the exponential distribution is given by: f(x) = λ * e^(-λx), where λ is the rate parameter of the distribution and e is the base of the natural logarithm.

In our case, the mean of the exponential distribution is given as 20 days. The rate parameter (λ) can be calculated as the reciprocal of the mean: λ = 1 / 20 = 0.05.

a) To find the probability that the time between two unplanned shutdowns is less than 14 days, we need to calculate the cumulative distribution function (CDF) at 14 days. The CDF gives the probability that the random variable is less than or equal to a certain value.

The CDF of the exponential distribution is given by: F(x) = 1 - e^(-λx).

Plugging in the values, we have: F(14) = 1 - e^(-0.05 * 14) ≈ 1 - e^(-0.7) ≈ 1 - 0.4966 ≈ 0.5034.

So, the probability that the time between two unplanned shutdowns is less than 14 days is approximately 0.5034, or 50.34%.

b) To find the probability that the time between two unplanned shutdowns is more than 21 days, we can calculate 1 minus the CDF at 21 days.

F(21) = 1 - e^(-0.05 * 21) ≈ 1 - e^(-1.05) ≈ 1 - 0.3499 ≈ 0.6501.

So, the probability that the time between two unplanned shutdowns is more than 21 days is approximately 0.6501, or 65.01%.

c) To find the probability that the time between two unplanned shutdowns is less than 7 days, we can directly calculate the CDF at 7 days.

F(7) = 1 - e^(-0.05 * 7) ≈ 1 - e^(-0.35) ≈ 1 - 0.7052 ≈ 0.2948.

So, the probability that the time between two unplanned shutdowns is less than 7 days is approximately 0.2948, or 29.48%.

Please note that these probabilities represent theoretical calculations based on the exponential distribution assumption and may not perfectly match real-world scenarios.