Question

Suppose you are part of the analytics team for the online retailer Macha Bucks which sells two types of tea to its online visitors: Rouge Roma (RR) and Emerald Earl (EE). Everyday approximately 10,000 people visit the site over a 24 hour period. For simplicity suppose we consider the “buy one or don’t buy” (BODB) market segment of customers which when they visit the site will conduct one of the following actions: (a) buy one order of RR, (b) buy one order of EE, or (c) don’t buy (DB) anything. You have been tasked with determining customer behavior on the website for the BODB segment using a random sample of 35 visits.

In the dataset for the random sample, each row corresponds to a random visitor. For each visitor we provide both the visitor’s action as well as the profit earned on the transaction. In the action column:

if the visitor buys one order of RR, we see a RR,
if the visitor buys one order of EE, we see an EE,
if the visitor doesn’t buy anything, we see a DB.

Note that even if two customers buy the same product, the profit can differ due to the shipping costs, promotions, or coupons that are applied

Random Sample of Data

1=yes, 0 = no

Transaction ID

Action

Profit ($)

Bought RR?

Bought EE?

Didn't Buy?

Profit RR ($)

Profit EE ($)

1

RR

8.43

1

0

0

.

0.00

2

DB

0.00

0

0

1

0.00

0.00

3

EE

1.75

0

1

0

0.00

1.75

4

DB

0.00

0

0

1

0.00

0.00

5

EE

4.37

0

1

0

0.00

4.37

6

EE

5.79

0

1

0

0.00

5.79

7

RR

6.27

1

0

0

6.27

0.00

8

RR

6.22

1

0

0

6.22

0.00

9

DB

0.00

0

0

1

0.00

0.00

10

EE

4.49

0

1

0

0.00

4.49

11

RR

10.54

1

0

0

10.54

0.00

12

EE

3.79

0

1

0

0.00

3.79

13

DB

0.00

0

0

1

0.00

0.00

14

DB

0.00

0

0

1

0.00

0.00

15

RR

9.03

1

0

0

9.03

0.00

16

EE

3.54

0

1

0

0.00

3.54

17

DB

0.00

0

0

1

0.00

0.00

18

DB

0.00

0

0

1

0.00

0.00

19

EE

5.02

0

1

0

0.00

5.02

20

DB

0.00

0

0

1

0.00

0.00

21

EE

3.60

0

1

0

0.00

3.60

22

DB

0.00

0

0

1

0.00

0.00

23

EE

2.61

0

1

0

0.00

2.61

24

RR

11.75

1

0

0

11.75

0.00

25

RR

12.22

1

0

0

12.22

0.00

26

DB

0.00

0

0

1

0.00

0.00

27

DB

0.00

0

0

1

0.00

0.00

28

EE

6.17

0

1

0

0.00

6.17

29

RR

8.83

1

0

0

8.83

0.00

30

DB

0.00

0

0

1

0.00

0.00

31

DB

0.00

0

0

1

0.00

0.00

32

DB

0.00

0

0

1

0.00

0.00

33

DB

0.00

0

0

1

0.00

0.00

34

RR

14.16

1

0

0

14.16

0.00

35

EE

6.06

0

1

0

0.00

6.06

PARTS

a) What could be an appropriate probability distribution to use for modeling the number of visitors that the website has in an hour?

b) What parameters would you use for the probability distribution?


c) Using that distribution, determine the probability that more than 600 people visit the site in an hour.

a) What could be an appropriate probability distribution to use for modeling the number of seconds between customer visits?

b) What parameters would you use for the probability distribution?


c) Using that distribution, determine the probability that the time between customer visits to the website is less than 10 seconds.

a) What could be an appropriate probability distribution to use for modeling the number of website visitors from 100 visitors that do not buy anything?

b) What parameters would you use for the probability distribution?


c) Using that distribution, determine the probability that from among 100 customers, it turns out that 30 or more customers do not buy anything.

d) What is the average number of visitors (from among 100 customers) that do not buy anything?

e) What is the standard deviation of the number of visitors (from among 100 customers) that do not buy anything?

What is the average profit from among 100 random customers that visit the site?
Please explain your answer or show your calculations.

Answer 1

a. The appropriate probability distribution for modelling the number of visitors that website has in one hour period is poisson distribution

b.the parameter we use for the poisson distribution is = mean no.of visitors that website has per hour

c.probability that more than 600 customers visit the site in an hour

from the given information we observe that website recieves 10000 visitors in 24 hours

Mean no .of visitors per hour = = 10000/24 = 416.666/ hour

now probability that more than600 customers visit the website per hour= P(X>600) = 1- P( X 600)

= 1- 1 =0

since by poisson distribution P( X 600) =1 if = 416.66

a.Appropriate probability distribution use for modelling no.of seconds between customer visits is Exponential distribution

since inter arrival time follows exponential distribution

b. the parameter for the inter arrival time distribution ( exponential distribution) is 1/

c. From the given data we observe that intrarrival time =1/= 1/416.666 = 0.0024 hours =8.64 seconds

.probability that the time between customer visits to the website is less than 10 seconds=

P ( t <10 seconds)= 0.9272 by exponential probability function with parameter =8.64

a. Appropriate probability distribution for modelling no of website customers from 100 visitors that do not buy any thing is Binomial distribution since the slogan of the website is buy one or dont buy ( i.e two possabilities i.e buy or dont buy

b.parameters of that distribution( binomial distribution) are n and p

n= no.of customers visited =100

p= proportion of customers do not buy

c.from the given data we observe that out of 35 customers visited the site 15 donot buy any thing

it means proportion of customers do not buy any thing = p=15/35=0.4286

proportion of customers buy one thing = q= 1- p=0.5714

now out of 100 customers 30 0r more do not buy any thing and its probability is given as

P(X >= 30) = 1- P(X<30)= 0.9969 from binomial probability function with parameters n=100 p=0.4286

hense probabilty that 0ut of 100 customers 30 or more do not buy any thing = 0.9969

Average profit from among 100 customers that visit the site

From the given data we observe that n= noof customers visited the web site = 35

No.of customers do not buy any thing =15

No.of customers buy one thing either EE or RR 20

Total profit from 35 customers visited the website = 134.64 USD

Now if 100 customers visit the website the total profit from them =384.69 USD

Average profit from 100 customers = 384.69/100 = 3.9469 USD / customer

This is very interesting case good experience