Answer:
a) 0.25
b) 52.76% probability that a person waits for less than 3 minutes
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
![f(x) = \lambda e^{-\lambda x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Clambda%20e%5E%7B-%5Clambda%20x%7D)
In which
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](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%20%5Cint%5Climits%5Ea_0%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Which has the following solution:
![P(X \leq x) = 1 - e^{-\mu x}](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%201%20-%20e%5E%7B-%5Cmu%20x%7D)
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}](https://tex.z-dn.net/?f=P%28X%20%3E%20x%29%20%3D%201%20-%20P%28X%20%5Cleq%20x%29%20%3D%201%20-%20%281%20-%20e%5E%7B-%5Cmu%20x%7D%29%20%3D%20e%5E%7B-%5Cmu%20x%7D)
In this question:
![m = 4](https://tex.z-dn.net/?f=m%20%3D%204)
a. Find the value of λ.
![\lambda = \frac{1}{m} = \frac{1}{4} = 0.25](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%3D%200.25)
b. What is the probability that a person waits for less than 3 minutes?
![P(X \leq 3) = 1 - e^{-0.25*3} = 0.5276](https://tex.z-dn.net/?f=P%28X%20%5Cleq%203%29%20%3D%201%20-%20e%5E%7B-0.25%2A3%7D%20%3D%200.5276)
52.76% probability that a person waits for less than 3 minutes