A sensor connected to a control center indicates that there is damage somewhere along a particular 4-mile length of train track. The conductor thi 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) Let f(x) be the probability density function of X. What is the value of f(1)? b) What is the expected value of X? c) What is the probability that the damage is located between mile 0 and mile 10 of that length of track?

Question

A sensor connected to a control center indicates that there is damage somewhere along a particular 4-mile length of train track.   The conductor thi 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) Let f(x) be the probability density function of X. What is the value of f(1)?   b) What is the expected value of X?   c) What is the probability that the damage is located between mile 0 and mile 10 of that length of track?

Anot1954 · Accepted Answer

Step 1 From the given information, the random variable X follows Uniform distribution with minimum value 0 and maximum value 4. f(x)=14−0=14,0&amp;lt;x&amp;lt;4 a) For continuous distributions, the probability at a particular point is zero. Here, the Uniform distribution is continuous distribution. Hence, f(1)=0 Step 2 b) Expected value of X: E(X)=a+b2=0+42=2 c) The probability that the damage is located between mile 0 and mile 10 of that length of track is, P(0&amp;lt;X&amp;lt;10)=∫010f(x)dx =∫04f(x)dx =∫0414dx =14(4−0) =1

A sensor connected to a control center indicates that there is damage

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