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Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at...

Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a service facility. Suppose they are normal random variables with means of 50 minutes, 60 minutes, and 40 minutes, respectively. The standard deviations are 15 minutes, 20 minutes, and 10 minutes, respectively.

a) Suppose X1, X2, and X3 are independent. All three repairs must be completed on a given object. What is the mean and variance of the total repair time for this object?

b) Suppose X1, X2, and X3 are independent. All three repairs must be completed on a given object. Find the probability that the total repair time is less than 180 minutes.

c) Suppose that X1, X2, and X3 are dependent so that the covariance between X1 and X2 is -150, between X1 and X3 is 60, and between X2 and X3 is -45. If all three repairs must be completed on a given object, what is the mean and variance of the total repair time for this object?

Solutions

Expert Solution

let Y represents the total repair time. Then,

, where,

given,

we are to find

[since independent]

Hence the required mean and variance of the total repair time of this object are

150 mins and 725 mins respectively.

we are to find

so,

[subtracting 150(mean of Y) on bothsides]

[dividing both sides by 26.93(SD of Y) on both sides]

[Z is a standard normal variable]

[value can be found from any normal table]

given,

we are to find

so,

[since dependent]

Hence the required mean and variance of the total repair time of this object are

150 mins and 455 mins respectively.

milcah answered 10 months ago

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