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Failures in machine in an 8 hrs shift is 1.5 failures. a) probability that in a...

Failures in machine in an 8 hrs shift is 1.5 failures.
a) probability that in a period of 16 hrs, exactly 2 failures take place.
b) probability that number of failures in 8 hrs exceed the media (medium value) in more than 1.5 standard deviations
Math   Statistics-And-Probability   
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Answer #1

Let X: failure in machines in an 8 hrs shift have exponential distribution with 1.5 failures.

a) P(X = 2)

For 16 hours the failure rate is 1.5 * 2 = 3 failures that is

The pdf of exponential distribution is,

The probability that exactly 2 failures takes place is 0.0074

b) P(X more than medium value)

The standard deviation of exponential distribution is,

Medium value is nothing but median.

The median of exponential distribution is,

The distribution function of exponential distribution is,

Probability that number of failures in 8 hours exceed the medium value is 0.422367

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