sensor connected to a control center indicates that there is damage somewhere along a particular 4-mile length of train track. The conductor thinks that the exact location of the damage is uniformly distributed between mile 0 and mile 4 of that length of track. Let X be the distance along the track from mile 0 to the location of the damage. a) What is the expected value of X? b) What is the probability that the damage is located between mile 2 and mile 4 of that length of track? c) Suppose the interval of track between miles 0 and 1 is checked carefully, and no damage is found. Given this, what is the probability that the damage is located betwee

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A sensor connected to a control center indicates that there is damage somewhere along a particular 4-mile length of train track.

The conductor thinks that the exact location of the damage is uniformly distributed between mile 0 and mile 4 of that length of track. Let be the distance along the track from mile 0 to the location of the damage.

a) What is the expected value of X?

b) What is the probability that the damage is located between mile 2 and mile 4 of that length of track?

c) Suppose the interval of track between miles 0 and 1 is checked carefully, and no damage is found. Given this, what is the probability that the damage is located between miles 2 and 4?

Expert Solution
Step 1

Given that X follows uniform distribution with minimum value 0 and maximum value 4.

a)

Expected value of X:

E(X)=a+b2=0+42=2

Step 2

b)

The probability that the damage is located between mile 2 and mile 4 of that length of track is,

P2<X<4=24f(x)dx                    =2414dx                   =14x24                   =144-2                  =0.5

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