Suppose that wait times for customers at a grocery store cashier line are uniformly distributed between one minute and twelve minutes.

a. What are the mean and variance of the waiting time?
b. What is the probability that a customer waits less than seven minutes?
c. What is the probability that a customer waits between four and twenty minutes?
d. Suppose that a customer who waits k minutes in line receives a coupon worth a 0.2 k^(1/4) dollar discount on a future visit. What is the mean of the coupon value for a customer?

a) Mean: 6.5 min, variance: 10.1 min

b) 0.54 (54%)

c) 0.73 (73%)

d) 0.34

Step-by-step explanation:

a)

Here we can call X the variable indicating the waiting time for the customers:

X = waiting time

We are told that the waiting time is distributed uniformly between 1 and 12; this means that

And the probability is equal for each value of X, so:

The mean of a uniform distribution is given by:

where a and b are the minimum and maximum values of the variable X. In this case,

a = 1

b = 12

So the mean value of X is

(minutes)

The variance of a uniform distribution is given by:

And substituting the values of this problem,

(minutes)

b)

Since the distribution is uniform between 1 and 12, we can write the probability density function as

The cumulative function gives the probability that the values of X is less than a certain value t:

(1)

In this case, we want to find the probability that the waiting time is less than 7 minutes, so

t = 7

We also have:

a = 1

b = 12

Therefore, calculating (1) and substituting, we find:

c)

The probability that a customer waits between four and twenty minutes can be rewritten as

This can be written as:

(1)

However, the probabilty of X>4 can be written as

Also, we notice that

because the maximum value of X is 12; therefore, we can rewrite (1) as

We can calculate by using the same method as in part b:

So, we find

d)

In this part, we know that a customer waits for

X = k

minutes in line, and he receives a coupon worth

dollars.

Here we want to find the mean of the coupon value.

Here therefore we have a new variables defined as

Given a variable with standard (between 0 and 1) uniform distribution X, the variable

follows a beta distribution, with parameters , and whose mean value is given by

In this case,

So the mean value of is

However, our variable is distribution is non-standard, because its values are between 1 and 12, so the range is

So, the actual mean value of is

However, in the  definition of Y we also have a factor 0.2; therefore, the mean value of Y is

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step-by-step explanation: umm..