Lecture#07-08.pdf - Contents VI 4 Multivariate Random...

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Contents VI Multivariate Random Variables 1 4 Definitions 1 4.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.2.1 Discrete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.2.2 Continuous Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Part VI Multivariate Random Variables 4 Definitions So far, we have dealt with univariate , or single-valued, random variables. We observe the outcome of one random experiment and then map the result to one real number. In many circumstances, it will become necessary to map the result of the random experiment to multiple random numbers. For example, Suppose we observe one apple tree during the growing season and we count Y 1 = the number of apples it produces and Y 2 = the number of mites observed on a random subset of 20 leaves. Both Y 1 and Y 2 are discrete random variables that map the outcome of a random experiment (of a tree growing through its growing season) to the real number line. Suppose I collect the exam score of 23 individuals ( X 1 , X 2 , . . . , X 23 ) by giving the same exam to all individuals in the class. In both cases, we may be interested in learning the relationships between the random numbers. These rela- tionships are unpredictable in detail because of the randomness, but predictable in trends. For example, the count of apples Y 1 may decline as the number of mites Y 2 increases. Also, scores may tend to increase to- gether if lectures were especially clear, or decrease together if the instructor assigned no homeworks covering the material. In what follows, we will focus on bivariate random variables ( Y 1 , Y 2 ) , but the definitions and results can extend to arbitrary n -variate random variables ( Y 1 , Y 2 , . . . , Y n ) . Definition : multivariate probability mass function (pmf) For discrete random variables ( Y 1 , Y 2 ) , the multivariate probability mass function (pmf) is de- fined as p ( y 1 , y 2 ) = P ( Y 1 = y 1 , Y 2 = y 2 ) the joint probability of events { Y 1 = y 1 } and { Y 2 = y 2 } . Definition : multivariate cumulative distribution function (cdf) For any random variables ( Y 1 , Y 2 ) , the multivariate cumulative distribution function (cdf) is defined as F ( y 1 , y 2 ) = P ( Y 1 y 1 , Y 2 y 2 ) , for all y 1 , y 2 ∈ <
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