Question

A study was conducted on the critical-part failures from 36 NASCAR races. The researchers discovered that the time in hours until the first critical-part failure is exponentially distributed with a mean of 0.5 hour. Find the probability that the time until the first critical-part failure is more than 2 hours.

Answer 1

Answer:

1.83% probability that the time until the first critical-part failure is more than 2 hours.

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

$f(x) = \mu e^{-\mu x}$

In which $\mu = \frac{1}{m}$ is the decay parameter.

The probability that x is lower or equal to a is given by:

$P(X \leq x) = \int\limits^a_0 {f(x)} \, dx$

Which has the following solution:

$P(X \leq x) = 1 - e^{-\mu x}$

The probability of finding a value higher than x is:

$P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}$

In this question, we have that:

$m = 0.5, \mu = \frac{1}{0.5} = 2$

Find the probability that the time until the first critical-part failure is more than 2 hours.

$P(X > 2) = e^{-2*2} = 0.0183$

1.83% probability that the time until the first critical-part failure is more than 2 hours.