Respuesta :
Answer:
a) 50.34% probability that the arrival time between customers will be 7 minutes or less.
b) 24.42% probability that the arrival time between customers will be between 3 and 7 minutes
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]
Mean of 10 minutes:
This means that [tex]m = 10, \mu = \frac{1}{10} = 0.1[/tex]
A. What is the probability that the arrival time between customers will be 7 minutes or less?
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
[tex]P(X \leq 7) = 1 - e^{-0.1*7} = 0.5034[/tex]
50.34% probability that the arrival time between customers will be 7 minutes or less.
B. What is the probability that the arrival time between customers will be between 3 and 7 minutes?
[tex]P(3 \leq X \leq 7) = P(X \leq 7) - P(X \leq 3)[/tex]
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
[tex]P(X \leq 7) = 1 - e^{-0.1*7} = 0.5034[/tex]
[tex]P(X \leq 3) = 1 - e^{-0.1*3} = 0.2592[/tex]
[tex]P(3 \leq X \leq 7) = P(X \leq 7) - P(X \leq 3) = 0.5034 - 0.2592 = 0.2442[/tex]
24.42% probability that the arrival time between customers will be between 3 and 7 minutes
