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Answer :
The average time in hours from when a machine breaks until it is fixed is approximately 11.3 hours.
Given,
Benny's arcade has five video game machines
The average time between failures = 34 hours
The maintenance engineer can repair a machine in about = 13 hours
The failure time and repair times are both exponentially distributed
The formula for the mean of an exponential distribution is mean = 1/λ where λ is the rate parameter of the distribution.
In this problem, the rate parameter of both the failure time and repair time is the reciprocal of their respective averages. Therefore,
λ_f = 1/34λ_r = 1/13
Now, let's find the average time from when a machine breaks until it is fixed using the fact that the sum of two independent exponential distributions with rate parameters λ1 and λ2 is itself an exponential distribution with rate parameter λ1+λ2.
So, the rate parameter for the time from when a machine breaks until it is fixed is λ_f+λ_r = 1/34+1/13
= 0.0885 hours⁻¹ (approximately)
Hence, the average time from when a machine breaks until it is fixed is mean = 1/λ= 1/0.0885 ≈ 11.3 hours (approximately).
Therefore, the average time in hours from when a machine breaks until it is fixed is approximately 11.3 hours.
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