Question

In a store, 40% of customers make a single purchase. This activity requires a time that has an exponential distribution with mean 2.0 minutes. The other 60% of customers ask for information before making a purchase. This process requires time and has a symmetric triangular distribution with between 1 and 5 minutes (in addition to the purchase time). Use Bernoulli, exponential and triangular random variates to generate a sample of shopping times for 200 customers. Plot the histogram of these observations.

Can you show steps in excel or what should I put in the column?

Answer 1

40% of the customers make a single purchase and 60% of the customers ask for information before making a purchase.

This is a Bernoulli distribution with probability p=0.40 that the customers make a single purchase

steps to simulate this

• Generate a random number from uniform distribution in the interval (0,1) using RAND()
• If the random number is less than 0.40 then the customer makes a single purchase, else the customer asks for information before making a purchase

The single purchase requires a time that has an exponential distribution with mean 2.0 minutes.

Let X be the time required for a single purchase. X has exponential distribution with parameter

The pdf of X is

The cdf of X is

We will get the inverse of this cdf using

Using the inverse cdf method we use the following steps to simulate the  time required for a single purchase.

• Generate a random number u from uniform distribution in the interval (0,1) using RAND()
• is a simulated value for the  time required for a single purchase.

The time required to ask for information before making a purchase that has has a symmetric triangular distribution with between 1 and 5 minutes. Since this is symmetric, the most likely value is (5+1)/2=3

Let Y be the time required to ask for information before making a purchase . Y has a triangular distribution with parameters a=1,b=5 and c=3

The pdf of Y is

The CDF of Y is

The inverse of cdf for 1<x<=3 is

The values of u for 1<x<=3 is 0<u<1/2

The inverse of cdf for 3<x<=5 is

The values of u for 3<x<=5 is 1/2<u<1

The steps for generating the time required to ask for information before making a purchase are

• Generate a random number u from uniform distribution in the interval (0,1) using RAND()
• if u is less than 1/2 then else

We prepare the following sheet

Copy the rows to create 200 trials. Paste as values to avoid changes

Get this

get the histogram of the total time using

data--->data analysis--->histogram

get this

format as needed