lecturenotesweek12 - ISyE 2027 Probability with...

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ISyE 2027 Probability with Applications Polly B. He Fall10, Week 12 Reading: Section 9.5, 10.1-10.2 Propagation of Independence Recall the independence of random variables. Question: Is independence of random variables preserved by functions? Example 9.5.1 Suppose X and Y are two independent random variables with joint distribu- tion function F . Define U = 1 { X ( a,b ] } and V = 1 { Y ( a,b ] } . Are U and V independent? How would you check? The general rule is called Propagation of Independence . Let X 1 , X 2 , . . . , X n be independent random variables. For each i , let h i : R R be a function and define the random variable Y i = h i ( X i ) . Then Y 1 , Y 2 , . . . , Y n are independent. Change-of-Variable Formula (Two-dimensional) Recall the one-dimensional change-of-variable formula: E[ g ( X )] = i g ( a i ) P ( X = a i ) if X is discrete. What about the two-dimensional case? Example 10.1.1 Suppose X is the number of salary raises for a company in a 2-year period and Y is the percentage of the raise(s). (Here we assume that once the company announces the %, it will stay the same for two years.) They have the joint p.m.f. as follows:
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