Question

1) In Australia, the time to repair a damaged automobile at a Smash Repair Shoppe has an distribution with an average time of 52.5 hours. (Note in the U.S. a Smash Repair Shoppe is called an Automobile Body Repair Shop.) A) What is the probability that a randomly selected damaged automobile will be repaired between 48.7 and 53.1 hours? B) Determine the probability that it takes less than the average time to repair a randomly selected damaged automobile.

Answer 1

Let X denotes the time to repair a damaged automobile.

X follows exponential distribution with mean 52.5 hours.

The pdf of X is

$f_X(x) = \frac{1}{52.5}e^{-\frac{x}{52.5}},x>0$

The cdf of X is

$F(x) =1-e^{-\frac{x}{52.5}},x>0$

a) $P(48.7 \leq X\leq 53.1)= F(53.1) - F(48.7)=[1-e^{-\frac{53.1}{52.5}}] - [1-e^{-\frac{48.7}{52.5}}]$

$\Rightarrow P(48.7 \leq X\leq 53.1)= e^{-\frac{48.7}{52.5}}-e^{-\frac{53.1}{52.5}}$

$\Rightarrow P(48.7 \leq X\leq 53.1)= 0.0317952$

b) $P(X\leq 52.5)= 1 - e^{-\frac{52.5}{52.5}} = 1-e^{-1} = 0.632121$