The time to failure, t, in hours, of a machine is often exponentially distributed with a probability density function f(t) = k e^(-kt), 0 to infinity, where k= 1/a and a is the average amount of time that will pass before a failure occurs. suppose that the average amount of time that will pass before a failure occurs is 80 hr. what is the probability that a failure will occur in 55 hr or less?

The probability is 0.4972

Step-by-step explanation:

The probability density function for the time of failure is:

f(t) = k e^(-kt)

Where k=1/a and a = 80hr so f(t) is equal to:

Then, if the probability density function follow a exponential distribution, the probability distribution function is:

The probability distribution function give as the probability that a failure will occur in t hours or less, so the probability that a failure will occur in 55 hr or less is calculated as:

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