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Unformatted text preview: Chapter 4: Pairs of Random Variables 4. Introduction This chapter analyzes experiments that produce two random variables, X and Y . The probability model for such an experiment contains the properties of the individual random variables and it also contains the relationship between the random variables. Chapter 5 analyzes the general case of experiments that produce n random variables. Twodimensional sample spaces result from experiments in which the interest is in the joint or simultaneous outcomes of two random variables. Pairs of random variables appear in a wide variety of practical situations. For example, we might be interested in the running speed and vertical leaping ability of an athlete, the signal emitted by a a radio transmitter and the corresponding signal that eventually arrives at a receiver, or in the temperature and volume of a gas. However, to be able to make probability statements about the joint outcomes of such random variables, we need to have information concerning their joint probability distribution. Although F X,Y ( x,y ), the joint cumulative distribution function of two random variables, is a complete probability model for any experiment that produces two random variables, it is not very useful for analyzing practical experiments. The more useful models are P X,Y ( x,y ), the joint probability mass function for two discrete random variables, and f X,Y ( x,y ), the joint probability density function of two continuous random variables. This chapter will consider both of the these joint probability models. 4.1 Joint Cumulative Distribution Function In the case of onedimensional sample spaces, that is, in experiments that produce single random variables, the events within these sample spaces represent points (in the discrete case) or intervals (in the continuous case) on a line. In the case of twodimensional sample spaces, that is, in experiments that lead to two random variables, X and Y , each outcome ( x,y ) is a point in a plane and events are points or areas in the plane. The joint CDF F X,Y ( x,y ) of two random variables is the probability of the area in the plane below and to the left of ( x,y ). Definition 4.1 Joint Cumulative Distribution Function (CDF) : The joint cumulative dis tribution function of random variables X and Y is F X,Y ( x,y ) = P [ X ≤ x,Y ≤ y ] . The joint CDF is a complete probability model. The subscripts of F are the names (in uppercase letters) of the two random variables; the arguments of the function that are associated with the random variable names are usually written as lowercase letters. Note that the event { X ≤ x } implies that Y can have any value so long as the condition on X is met. Therefore, F X ( x ) = P [ X ≤ x ] = P [ X ≤ x,Y < ∞ ] = P lim y →∞ [ X ≤ x,Y ≤ y ] 59 = lim y →∞ P [ X ≤ x,Y ≤ y ] = lim y →∞ F X,Y ( x,y ) = F X,Y ( x, ∞ ) ....
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This document was uploaded on 10/28/2010.
 Spring '09

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