Final answer:
a) The mean and variance of the distance between successive flaws on the wire is 0.67 meters. b) The probability that the distance between a randomly selected flaw and the next flaw is at least 1 meter is 0.2231. c) The probability that the distance is no more than 0.2 meter is 0.1813. d) The probability that the distance is between 0.5 and 1.5 meters is 0.3126.
Step-by-step explanation:
a) Mean and Variance of Distance between Successive Flaws
The mean distance between successive flaws on the wire can be found using the formula: mean = 1/λ, where λ is the rate of the Poisson process. In this case, the rate is 1.5 flaws per meter, so the mean distance is 1/1.5 = 0.67 meters.
The variance of the distance between successive flaws is equal to the mean distance, so the variance is also 0.67 meters.
b) Probability of Distance >= 1 Meter
The distance between two successive flaws follows an exponential distribution with parameter λ. The probability that the distance is at least 1 meter can be found using the formula: P(X >= x) = e^(-λx), where x is the distance. Plugging in the values, we get P(X >= 1) = e^(-1.5*1) = 0.2231.
c) Probability of Distance <= 0.2 Meter
The probability that the distance is no more than 0.2 meter can be found using the formula: P(X <= x) = 1 - e^(-λx). Plugging in the values, we get P(X <= 0.2) = 1 - e^(-1.5*0.2) = 0.1813.
d) Probability of Distance between 0.5 and 1.5 Meters
The probability that the distance is between 0.5 and 1.5 meters can be found by subtracting the cumulative probabilities: P(0.5 <= X <= 1.5) = P(X <= 1.5) - P(X <= 0.5) = (1 - e^(-1.5*1.5)) - (1 - e^(-1.5*0.5)) = 0.3126.