In a manufacturing process involving several machines, the average down time t (in hours) for a machine that needs repair has the probability density function . Find the probability that a failed machine's down time is 4 hours or more. Round your answer to three decimal places.

Question

In a manufacturing process involving several machines, the average down time t (in hours) for a machine that needs repair has the probability density function  . Find the probability that a failed machine's down time is 4 hours or more. Round your answer to three decimal places. ​

Quizplus · Accepted Answer

We need to find P(t ≥ 4), which is equivalent to 1 - P(t < 4). Using the given probability density function, we can integrate it from 0 to 4 to get the probability of a failed machine having a down time of less than 4 hours:
P(t < 4) = ∫₀ ⁴λe^(-λt) dt = (1 - e^(-4λ))/λ
To find λ, we use the fact that the integral of a probability density function over its entire domain (in this case, from 0 to infinity) is equal to 1. 
∫₀   ∞λe^(-λt) dt = 1
Solving for λ, we get λ = 0.25.
Plugging in λ = 0.25 into the equation for P(t < 4), we get:
P(t < 4) = (1 - e^(-1))/0.25 ≈ 0.865
Therefore, the probability that a failed machine's down time is 4 hours or more is approximately 0.135.

In a Manufacturing Process Involving Several Machines, the Average Down