Question

The wait time at the Goleta Post Office is uniformly distributed between 1 and 16 minutes.

a) Define the random variable of interest, X.

b) State the distribution of X.

c) What is the average wait time?

d) Calculate the probability that the wait time is more than 17 minutes.

e) Calculate the probability that the wait time is at least 10 minutes.

f) Calculate the probability that the wait time is between 2 and 11 minutes

Answer 1

Uniform Distribution

A continuous random variable X is said to have a uniform distribution over (a,b) if the PDF(Probability Distribution Function) is given by,

Notation: X~Uniform(a,b)

The average or the mean of the uniform distribution is given by,

where E(X) is the expectation of X.

Coming back to our problem,

Given that the wait time at the Goleta Post Office is uniformly distributed between 1 and 16 minutes.

a) Here we need define the random variable of interest X i.e.

X=Wait time at the Goleto Post Office

b) Here we need to state the distribution of X,

Clearly X~Uniform(a=1,b=16)

The PDF of X is given by,

c) Here we need to find the average wait time,

Hence the average wait time is 8.5.

d) Here we need to find the probability that the wait time is more than 17 minutes,

Hence the probability that the wait time is more than 17 minutes is 0.

e) Here we need to find the probability that the wait time is atleast 10 minutes,

Hence the probability that the wait time is atleast 10 minutes is 0.6.

f) Here we need to find the probability that the wait time is between 2 and 11 minutes,

Hence the probability that the wait time is between 2 and 11 minutes is 0.6.