Air quality monitoring stations are maintained throughout Maricopa County, Arizona and the Phoenix metropolitan area. Measurements for particulate matter and ozone are measured hourly. Particulate matter (known as PM10) is a measure (in μg/m3) of solid and liquid particles in the air with diameters less than 10 micrometers. Ozone is a colorless gas with molecules comprised of three oxygen atoms that make it very reactive. Ozone is formed in a complex reaction from heat, sunlight, and other pollutants, especially volatile organic compounds. The U.S. Environmental Protection Agency sets limits for both PM10 and ozone. For example, the limit for ozone is 0.075 ppm. The probability a day in Phoenix exceeds the limits for PM10 and ozone is important for compliance and remedial actions with the county and city. But this might be more involved that the product of the probabilities for each pollutant separately. It might be that days with high PM10 measurements also tend to have ozone values. That is, the measurements might not be independent and it is the joint relationship between these measurements that becomes important. The study of probability distributions for more than one random variable is the focus of this chapter and the air quality data is just one illustration of the ubiquitous need to study variables jointly.

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