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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen JanuaryApril 2011 Jiahua Chen () Lecture 10 JanuaryApril 2011 1 / 23 Disk example Consider a random experiment in which a dart is thrown randomly to disk of radius 1. Assume each point on the disk is equally likely hit. In a standard coordinate system, the sample space is presented as Ω = { ( x , y ) : x 1 + y 2 ≤ 1 } . Any subset of Ω , say A , has its probability defined as P ( A ) = area of A area of Ω . Jiahua Chen () Lecture 10 JanuaryApril 2011 2 / 23 Two random variables, joint random behaviour Define two random variables, X and Y to be the xy coordinates of the landing point on the disk. What is their joint distribution? That is, how much is P ( X ≤ − 0.5, Y ≤ 0.3 ) for all ( x , y ) such as (0.5, 0.3) as given above. The bivariate function F ( x , y ) = P ( X ≤ x , Y ≤ y ) is called the joint cumulative distribution function of X and Y . For the current example, F ( x , y ) is a continuous function. Hence, we say that X and Y are jointly continuous. Jiahua Chen () Lecture 10 JanuaryApril 2011 3 / 23 Joint cumulative distribution function The joint cumulative distribution function of X and Y is defined for any pair of random variables; discretediscrete, continuouscontiuous, discretecontinuous or others. It is seen that ≤ F ( x , y ) ≤ 1 F ( − ∞ , − ∞ ) = 0 and F ( ∞ , ∞ ) = 1. F ( x , y ) is increasing in both x , and y . More details will be omitted from this course. Jiahua Chen () Lecture 10 JanuaryApril 2011 4 / 23 Back to disk example Recall that X and Y are the xy coordinates of the landing point on the disk. We wish to compute P ( X ≤ − 0.5, Y ≤ 0.3 ) or P ( X ≤ x , Y ≤ y ) . The area formed by A = { ( x , y ) : x ≤ − 0.5, y ≤ 0.3, x 2 + y 2 ≤ 1 } can be computed via integration: integraldisplay 0.3 − √ 0.75 ( − 0.5 + radicalbig 1 − y 2 ) dy = 0.4525. Hence, we find P ( X ≤ − 0.5, Y ≤ 0.3 ) = 0.4525 π = 0.1440. Remark: a numerical error has been corrected. Jiahua Chen () Lecture 10 JanuaryApril 2011 5 / 23 Back to disk example It is apparent now that computing P ( X ≤ x , Y ≤ y ) is possible in terms of finding an analytical expression. Yet you are assured that the expression is not pleasant to present here nor for your eyes. Jiahua Chen () Lecture 10 JanuaryApril 2011 6 / 23 Back to disk example At the same time, we have P ( X ≤ x , Y ≤ y ) = integraldisplay y t = − ∞ integraldisplay x s = − ∞ f ( s , t ) dsdt with f ( x , y ) = 1 π I ( x 2 + y 2 ≤ 1 ) ....
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This note was uploaded on 04/07/2011 for the course STAT 302 taught by Professor Dr.chen during the Spring '11 term at The University of British Columbia.
 Spring '11
 Dr.Chen
 Probability

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