Question

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 1

given that

ref\stoves	0	1	2	total P(Y=y)
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
total P(X=x)	0.22	0.59	0.29	1

let X is number of stoves while Y is the number of refrigerators

b)

laws of discrete probability

c)

we have to find P(Y=2|X=2)

since from table we can see that P(X=2)=0.29

P(X=2,Y=2)=0.04

so

d)

e)

Var(Y)=E(Y²)-(E(Y))²=1.47-0.93²=0.6051

f)

we have to check if X and Y are independent or not

for that we need to check if P(X=x,Y=y)=P(X=x)*P(Y=y) for all then X and Y are independent otherwise not

hence

P(X=0,Y=0)=0.08

P(X=0)*P(X=0)=0.22*0.34=0.0748

since P(X=0,Y=0) not equal to P(X=0)*P(Y=0)

Hence X and Y are not independent

g)

we have to find P(Y=y|X=0) for y=0,1,2

Hence

h)

we have to find E(Y|X=0) and Var(Y|X=0)

now

=0.8637

=1.3183

so Var(Y|X=0) =E(Y²|X=0)-(E(Y|X=0))²=1.3183-0.8637²=0.5723