Answer:
6 students are served per hour.
45.12% probability a student waits less than 6 minutes.
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}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cmu%20e%5E%7B-%5Cmu%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)
mean of 10 minutes.
This means that
, so ![\mu = \frac{1}{10} = 0.1](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20%3D%200.1)
How many students are served per hour?
One student is served each 10 minutes, on average
An hour has 60 minutes
60/10 = 6
6 students are served per hour.
Calculate the probability a student waits less than 6 minutes.
![P(X \leq x) = 1 - e^{-0.1*6} = 0.4512](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%201%20-%20e%5E%7B-0.1%2A6%7D%20%3D%200.4512)
45.12% probability a student waits less than 6 minutes.