Suppose the average tread-life of a certain brand of tire is 42,000 miles and that this mileage follows the exponential probability distribution. What is the probability that a randomly selected tire will have a tread-life of less than 65,000 miles?

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Quizplus · Accepted Answer

The exponential probability distribution function is given by $f(x) = \lambda e^{-\lambda x}$, where $\lambda = \frac{1}{\mu}$ and $\mu$ is the mean of the distribution. For a tire with an average tread-life of 42,000 miles, $\lambda = \frac{1}{42000}$. The probability that a tire will have a tread-life of less than 65,000 miles is calculated using the cumulative distribution function (CDF), which is $1 - e^{-\lambda x}$. Substituting $\lambda = \frac{1}{42000}$ and $x = 65000$, we get $1 - e^{-\frac{65000}{42000}}$, which results in approximately 0.7872.

Suppose the Average Tread-Life of a Certain Brand of Tire