a. (5 pts) Create a (4 x 3) table that shows the joint probability mass function

Question

a. (5 pts) Create a (4 x 3) table that shows the joint probability mass function (PMF) of house vintage and renovation status. Use the letters (a, b, …, k, l) to identify the number of households with each possible joint value of the Vintage and Renovate random variables (i.e., the number of households in each cell).

b. (10 pts) Using your lettering system, what is the probability that:

i. A home has more than 50% of glass sq. footage having been replaced?

ii. A home that has had between 1% and 50% of glass replaced?

iii. A home has some glass sq. footage replaced?

c. (15 points) Augment the Table you created in (a) and using your notation:

i. Augment the Table you created in (a) that shows the marginal Vintage and Renovate percentages.

ii. Create a table that shows the conditional renovation status PMF for each house vintage

iii. Create a table that shows the conditional age status PMF for each degree of renovation.

iv. What is the probability that less than 25% of glass sq. footage has been replaced given that the randomly chosen house is an old home?

v. Given that a house has been identified has having more than 50% of glass replaced, what is the probability distribution of house age status?

vi. Given a new house, what is the probability of the house have no glass replaced or more than 50% of glass replaced?

1. (30 pts) Consider a sample of T households. Suppose you work for Wisconsin's Focus on Energy as a policy analyst. You are involved in a project examining the determinants of households undertaking specific home energy conservation home. In one analysis you want to examine window replacement in homes of different vintages. You have surveyed that above T households where you have information on: * House Age 0-0 Very Old: built prior to 1950 o = 1 Old: built between 1950 and 1975 0-2 New: built between 1975 and 2005] Degree to which the homeowner(s) have installed energy efficient windows 0-0-)None: 0% glass area 0-1 Limited: 1%-25% of glass area . 2 Significant: Between 25% and 50% of glass area Substantial: More than 50% of glass area renovated. 0-3 So in these T households there are (3 x 4) combinations of house vintage and window renovation status. Note that a particular household will be represented by only one of these combinations. We can define two discrete random variables, Vintage and Renovate for this population via the following:

Explanation / Answer

Solution

Part (1)

(4 x 3) table of joint probability mass function

Vinatge

Renovate

0

1

2

3

0

a/T

b/T

c/T

d/T

1

e/T

f/T

g/T

h/T

2

i/T

j/T

k/T

l/T

Part (2)

Probability that:a home has more than 50% of glass sq. footage having been replaced = (d + h + l)/T

Part (3)

Probability that:a home that has had between 1% and 50% of glass replaced = (b + c + f + g + j + k)/T}

Part (4)

Probability that:a home has some glass sq. footage replaced = 1 – (a + e + i)T

Part (5) Augmented Table

Vinatge

Renovate

Marginal

0

1

2

3

Percetage

0

a/T

b/T

c/T

d/T

(a + b+ c + d)T

1

e/T

f/T

g/T

h/T

(e+ f+ g + h)T

2

i/T

j/T

k/T

l/T

(i + j+ k + l)T

Marginal

Percentage

(a + e + i)/T

(b + f + j)/T

(c + g + k)/T

(d + h + l)/T

1

DONE

Vinatge

Renovate

0

1

2

3

0

a/T

b/T

c/T

d/T

1

e/T

f/T

g/T

h/T

2

i/T

j/T

k/T

l/T

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