Question

For a group of 100,000 homes, the table below shows the frequency distribution for the number of persons per occupied housing unit. Frequencies are on thousands of housing units.

Persons

1 2 3 4 5 6 7

Frequency

25.8 32.6 16.4 ? 6.2 2.3 1.4

The random variable Y gives the occupancy of a randomly selected home from this group.

a.Complete the following table.

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Y:

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Pr (Y):

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∑YPr(Y):

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∑Y^2Pr(Y) :

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b.Find the mean and variance of Y

c.Find the probability that Y is less than its mea.

d.Graph the pmf and cdf of Y

.

Answer 1

Since freqeuncies are on 1000 so sum of frequencies must be 100. To find the missing frequency we need to subtract sum of given freqeuncies from 100. Following is the completed table:

Persons	Frequency
1	25.8
2	32.6
3	16.4
4	15.3
5	6.2
6	2.3
7	1.4

Following table shows the completed table:

Persons	Frequency, f	Pr(Y=y)=f/100	yPr(Y=y)	y^2P(Y=y)
1	25.8	0.258	0.258	0.258
2	32.6	0.326	0.652	1.304
3	16.4	0.164	0.492	1.476
4	15.3	0.153	0.612	2.448
5	6.2	0.062	0.31	1.55
6	2.3	0.023	0.138	0.828
7	1.4	0.014	0.098	0.686
Total	100		2.56	8.55

(B)

The mean is:

The variance is:

(c)

(d)

The graph is:

Following table shows the CDF:

Persons	Frequency, f	Pr(Y=y)=f/100	P(Y<=y)
1	25.8	0.258	0.258
2	32.6	0.326	0.584
3	16.4	0.164	0.748
4	15.3	0.153	0.901
5	6.2	0.062	0.963
6	2.3	0.023	0.986
7	1.4	0.014	1
Total	100

Following is the graph: