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After analyzing several months of sales data, the owner of an appliance store produced the following joint probability distribution of the number of refrigerators and stoves sold hourly
0 1 2 Stoves
0 0.08 0.14 0.12 0.34
1 0.09 0.17 0.13 0.39
2 0.05 0.18 0.04 0.27
REF 0.22 0.49 0.29 1
b. What are the laws for a discrete probability density function?
c. If a customer purchases 2 stoves, what is the probability they will also purchase two refrigerators?
d. What is the average number of refrigerators purchased?
e. What is the variance in the number of refrigerators purchased?
f. Are the sale of stores and refrigerators independent?
g. What is the conditional probability distribution for sales in refrigerators if the customer did not purchase a stove?
h. What is the expected value and variance for sales in refrigerators, if the customer did not purchase a stove?

Answer :

Each probability lies between 0 to 1, If a customer purchases 2 stoves, the probability is 0.1379 they will also purchase two refrigerators, the average number of refrigerators purchased is 0.93, and the variance in the number of refrigerators purchased is 0.6051.

A)     0            1            2          stoves

0      0.08       0.14      0.12      0.34

1       0.09       0.17      0.13      0.39

2      0.05       0.18      0.04     0.27

RF    0.22       0.49     0.29     1

B) Each probability lies between 0 to 1

All the probabilities sum up to 1

C) P(2 Ref/ 2 stoves) = P(2 Ref and 2 stoves) / 2 stoves)

= 0.04/0.29 = 4/29 = 0.1379

D) Average number of Ref purchased E(x) = ∑xi pi

X        0          1         2

Prob.  0.34    0.39   0.27

= 0.39 + 0.54

= 0.93

E) Var(x) = ∈(x)² - (∈{x})²

= (0.39 + 1.08) - (0.93 * 0.93)

= 1.47 - 0.8649

= 0.6051

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