Answer :
Answer:
a) 0.426 = 42.6% probability that a customer waits less than a second for credit card approval.
b) 0.189 = 18.9% probability that a customer waits more than 3 seconds.
c) The minimum approval time for the slowest 5% of transactions is 5.39 seconds.
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]
An online retailer has determined that the average time for credit card transactions to be electronically approved is 1.8 seconds.
This means that [tex]m = 1.8, \mu = \frac{1}{1.8} = 0.5555[/tex]
(a) Use an exponential density function to find the probability that a customer waits less than a second for credit card approval.
This is [tex]P(X \leq 1)[/tex].
[tex]P(X \leq 1) = 1 - e^{-0.5555*1} = 0.426[/tex]
0.426 = 42.6% probability that a customer waits less than a second for credit card approval.
(b) Find the probability that a customer waits more than 3 seconds.
This is P(X > 3).
[tex]P(X > x) = 1 - P(X \leq 3) = 1 - (1 - e^{-0.5555*3}) = 0.189[/tex]
0.189 = 18.9% probability that a customer waits more than 3 seconds.
(c) What is the minimum approval time for the slowest 5% of transactions
This is x for which [tex]P(X \leq x) = 0.95[/tex]
So
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
[tex]0.95 = 1 - e^{-0.5555x}[/tex]
[tex]e^{-0.5555x} = 0.05[/tex]
[tex]\ln{e^{-0.5555x}} = \ln{0.05}[/tex]
[tex]-0.5555x = \ln{0.05}[/tex]
[tex]x = -\frac{\ln{0.05}}{0.5555}[/tex]
[tex]x = 5.39[/tex]
The minimum approval time for the slowest 5% of transactions is 5.39 seconds.