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5
Joint Probability
Distributions
Air-quality monitoring stations are maintained throughout
Chapter Outline Maricopa County, Arizona and the Phoenix metropolitan
area. Measurements for particulate matter and ozone are
5-1 Two or More Random Variables measured hourly. Particulate matter (known as PM10) is a
5-1.1 Joint Probability Distributions measure (in μg m/ 3 ) of solid and liquid particles in the air
5-1.2 Marginal Probability Distributions with diameters less than 10 micrometers. Ozone is a color-
5-1.3 Conditional Probability less gas with molecules comprised of three oxygen atoms
Distributions that make it very reactive. Ozone is formed in a complex
5-1.4 Independence reaction from heat, sunlight, and other pollutants, especially
5-1.5 More Than Two Random Variables volatile organic compounds. The U.S. Environmental Pro-
tection Agency sets limits for both PM10 and ozone. For
5-2 Covariance and Correlation example, the limit for ozone is 0.075 ppm. The probability
5-3 Common Joint Distributions that a day in Phoenix exceeds the limits for PM10 and ozone
5-3.1 Multinomial Probability is important for compliance and remedial actions with the
Distribution county and city. But this might be more involved that the
5-3.2 Bivariate Normal Distribution product of the probabilities for each pollutant separately.
It might be that days with high PM10 measurements also tend
5-4 Linear Functions of Random Variables to have ozone values. That is, the measurements might not be
independent, so the joint relationship between these meas-
5-5 General Functions of Random Variables urements becomes important. The study of probability dis-
tributions for more than one random variable is the focus of
5-6 Moment Generating Functions this chapter and the air-quality data is just one illustration
of the ubiquitous need to study variables jointly.
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