Question

Persons per Housing Unit. From the document American Housing Survey for the United States, published by the U.S. Census Bureau, we obtained the following frequency distribution for the number of persons per occupied housing unit, where we have used "7" in place of “7 or more.” Frequencies are in millions of housing units.

Person	1	2	3	4	5	6	7
Frequencies	27.9	34.4	17.0	15.5	6.8	2.3	1.4

For a randomly selected housing unit, let Y denote the number of persons living in that unit.

a. Identify the possible values of the random variable Y.

b. Use random-variable notation to represent the event that a housing unit has exactly three persons living in it.

c. Determine P(Y = 3); interpret in terms of percentages.

d. Determine the probability distribution of Y.

e. Construct a probability histogram for Y.

Short answer

Part a.

The possible values for Y: {1,2,3,4,5,6,7}

Part b.

Y = 3

Part c.

P(Y = 3) = 17/105.3 = 0.161

Part d.

Random Variable	Probability
1	27.9
2	34.4
3	17.0
4	15.5
5	6.8
6	2.3
7	1.4

Part e.

Step-by-step solution

Part (a) Step 1. Given information

The frequency distribution for the number of people per occupied dwelling unit is shown below, where "7" has been substituted for "7 or more." The following frequencies are found in millions of home units:

Persons	1	2	3	4	5	6	7
Frequency	27.9	34.4	17.0	15.5	6.8	2.3	1.4

Let Y represent the number of people drawn at random from a housing unit.

Part (a) Step 2. Solution

The number of people per occupied housing unit can be calculated using the frequency distribution table provided. be 1, 2, 3, 4, 5, 6 or 7. Hence, the potential values for Y:

$Y=1,2,3,4,5,6,7$

Part (a) Step 3. Solution

When a living unit has exactly three people, it is referred to as:

Y = 3

Part (c) Step 1. Formula Used 

The experiment is conducted out assuming that the random variable $(Y=y)$occurs n times out of a total of N times. As a result of the $\frac{f}{N}$rule:

$P(Y=y)=\frac{n}{N}$

Part (c) Step 2. Solution

Here, Total number of times the experiment took place (N):

$27.9+34.4+17+15.5+6.8+2.3+1.4=105.5$

By $\frac{f}{N}$rule:

$$\begin{gathered} P(Y=3)=\frac{17}{105.5} \\ =0.161 \end{gathered}$$

Part (d) Step 1. Solution

Here, total number of times the experiment took place :

$27.9+34.4+17+15.5+6.8+2.3+1.4=105.5$

Hence the probability of random variable Y is:

Random Variable	Probability
1	27.9
2	34.4
3	17.0
4	15.5
5	6.8
6	2.3
7	1.4

Part (e) Step 1. Solution

The probability histogram for the random variable X is as follows: