lecturenotesweek11

# lecturenotesweek11 - ISyE 2027 Probability with...

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ISyE 2027 Probability with Applications Polly B. He Fall10, Week 11 Reading: Section 9.4 Independent Random Variables Recall the independence of events. We say two events A and B are independent if P ( A B ) = P ( A ) P ( B ). What about the independence between random variables? Quick exercise: do you think the following random variables are independent? 1. We toss a fair coin once and then roll a die. Let X be 1 if the coin comes up as a head and 0 otherwise. Let Y be the outcome of the die. 2. The price movements of corn and soy beans on the Chicago Mercantile Exchange. Deﬁnition: Two random variables X and Y , with joint distribution function F , are indepen- dent if P ( X a,Y b ) = P ( X a ) P ( Y b ) , i.e., F ( a,b ) = F X ( a ) F Y ( b ) , for all -∞ < a,b < Further, checking the last equation is equivalent to checking one of the following statements: 1. For all sets A and B , P ( X A,Y B ) = P ( X A ) P ( Y B ) 2. If X and Y are both discrete random variables,

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lecturenotesweek11 - ISyE 2027 Probability with...

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