Answer :
Answer:
0.4968 = 49.68% probability that the repair time takes between 2 to 4 hours
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
The probability of finding a value higher than x is:
[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
Average repair time of 2 hours
This means that [tex]m = 2, \mu = \frac{1}{2} = 0.5[/tex]
What is the probability that the repair time takes between 2 to 4 hours
More than 2:
[tex]P(X > 2) = e^{-0.5*2} = 0.3679[/tex]
Less than 4:
[tex]P(X \leq 4) = 1 - e^{0.05*4} = 0.8647[/tex]
Between two and four:
[tex]P(2 \leq x \leq 4) = 0.8647 - 0.3679 = 0.4968[/tex]
0.4968 = 49.68% probability that the repair time takes between 2 to 4 hours