The U.S. Coast Guard (USCG) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents (USCG website, November 8, 2012).
Wind Condition | Percentage of Accidents |
None | 9.6 |
Light | 57.0 |
Moderate | 23.8 |
Strong | 7.7 |
Storm | 1.9 |
Let x be a random variable reflecting the known wind condition at the time of each accident. Set x = 0 for none, x = 1 for light, x = 2 for moderate, x = 3 for strong, and x = 4 for storm.
Comment on what your results imply about the wind conditions during boating accidents.
a.
Construct a probability distribution for the random variable x.
The probability distribution for the random variable x is given by,
x | |
0 | 0.0960 |
1 | 0.05700 |
2 | 0.2380 |
3 | 0.0770 |
4 | 0.0190 |
Calculation:
The data represents the results obtained for 4,401 boating accidents including the wind condition at the time of the accident. The random variable x represents the known wind condition at the time of each accident. The random variable x takes the value 0 for none,
takes the value 1 for light, takes the value 2 for moderate, takes the value 3 for strong, takes the value 4 for storm.
Here, the total number of responses is 4,401. The corresponding probabilities are obtained by converting the percentages in to probabilities. That is, by dividing each value with 100.
The probability distribution for the random variable x can be obtained as follows:
x | f | ||
0 | 9.6 | 0.0960 | |
1 | 57.0 | 0.5700 | |
2 | 23.8 | 0.2380 | |
3 | 7.7 | 0.0770 | |
4 | 1.9 | 0.0190 | |
Total | 100 | 1 |
b.
Find the expected value for the random variable x.
The expected value for the random variable x is 1.353.
Calculation:
The formula for the expected value of a discrete random variable is,
The expected value for the random variable x is obtained using the following table:
x | f(x) | |
0 | 0.096 | 0 |
1 | 0.57 | 0.57 |
2 | 0.238 | 0.476 |
3 | 0.077 | 0.231 |
4 | 0.019 | 0.076 |
Total | 1 | 1.353 |
Thus, the expected value for the random variable x is 1.353.
c.
Find the variance and standard deviation of the random variable x.
The variance of the random variable x is 0.6884.
The standard deviation of the random variable x is 0.8297.
Calculation:
The formula for the variance of the discrete random variable is,
The variance of the random variable x is obtained using the following table:
x | f(x) | |||
0 | 0.096 | –1.353 | 1.8306 | 0.1757 |
1 | 0.57 | –0.353 | 0.1246 | 0.0710 |
2 | 0.238 | 0.647 | 0.4186 | 0.0996 |
3 | 0.077 | 1.647 | 2.7126 | 0.2089 |
4 | 0.019 | 2.647 | 7.0066 | 0.1331 |
Total | 1 | 3.235 | 12.0930 | 0.6884 |
Therefore,
Thus, the variance of the random variable x is 0.6884.
The formula for the standard deviation of the discrete random variable is,
Thus, the standard deviation is,
Hence, the standard deviation of the random variable x is 0.8297.
d.
Explain what the result implies about the wind conditions during the boating accidents.
The expected value is 1.353 and it represents the mean wind conditions when accident occurs. This value is slightly less than light wind conditions.
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