Given Information:
Exponential distribution
Mean = μ = 12 seconds
Required Information:
Mean time between complaints = ?
P(X < mean) = ?
P(5 < X < 10) = ?
Explanation:
X = time of complaint against Britain’s biggest banks in 2011 has an exponential distribution with the average complaint time of 12 seconds.
Mean = μ = 12 seconds
Decay rate = λ = 1 /μ = 1/12 = 0.083
The standard deviation is equal to the mean in exponential distribution = σ = μ = 12
a. What is the mean time between complaints?
The mean time between complaints is 12 seconds
The probability of that happening is
P(X = 12) = 0.083e^–0.083x(12) = 0.030
b. What is the probability that the next complaint will take less than the mean time?
P(X < mean) = 1 – e^–0.083x
P(X < 12) = 1 – e^–0.083(12)
P(X < 12) = 1 – 0.369
P(X < 12) = 0.63 = 63%
There is 63% probability that the next complaint will take less than the mean time of 12 seconds.
c. What is the probability that the next complaint will take between 5 and 10 seconds?
P(5 < X < 10) = P(X < 10) – P(X < 5)
P(X < 10) = 1 – e^(–0.083)(10) = 0.563
P(X < 5) = 1 – e^(–0.083)(5) = 0.339
P(5 < X < 10) = 0.563 − 0.339 = 0.224
P(5 < X < 10) = 22.4 %
There is 22.4% probability that the next complaint will take between 5 and 10 seconds.