Question

The following data are collected to study unemployment in a particular region in upstate New York. The first column shows the number of months unemployed and the second column shows the count of people in the study with the corresponing unemployment length.

Let X be a random variable indicating the number of months a person is unemployed.

a)Use the data to develop the probability mass function for X.

b)Find the cumulative distribution function for X.

c)Calculate the mean and standard deviation of X.

Months Unemployed	Number Unemployed
1	1029
2	1686
3	2269
4	2675
5	3487
6	4652
7	4145
8	3587
9	2325
10	1120

Answer 1

a) For the given data, the sum of frequency is computed to be 26975. The probability for each value of months employed is computed by dividing the corresponding frequency with that number os 26975 to get the PMF here as:

Months Unemployed	Number Unemployed	P(X = x)	F(x) = P(X <= x)
1	1029	0.03814643	0.038146432
2	1686	0.06250232	0.100648749
3	2269	0.08411492	0.18476367
4	2675	0.09916589	0.283929564
5	3487	0.12926784	0.413197405
6	4652	0.17245598	0.585653383
7	4145	0.1536608	0.73931418
8	3587	0.13297498	0.872289157
9	2325	0.08619092	0.958480074
10	1120	0.04151993	1
		26975

The third column above represents the PMF For X here.

b) The CDF is computed by simply finding the cumulative sum of the PDF column to get the 4^th column in the above table.

The 4th column in the above table is the CDF for X here.

c) The mean of X here is computed as:

Therefore 5.8236 is the mean number of months unemployed here.

The second moment of X is first computed here as:

The standard deviation now is computed here as:

Therefore 2.3089 is the required standard deviation here.