Lecturenotes14

Lecturenotes14 - STAT 430/510 Lecture 14 STAT 430/510...

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STAT 430/510 Lecture 14 STAT 430/510 Probability Hui Nie Lecture 14 June 22nd, 2009
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STAT 430/510 Lecture 14 Independence Two events A and B are independent if P ( AB ) = P ( A ) P ( B ) Two random variables X and Y are independent if for any two sets of real numbers A and B , P ( X A , Y B ) = P ( X A ) P ( Y B ) This implies that for all a , b , P ( X a , Y b ) = P ( X a ) P ( Y b )
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STAT 430/510 Lecture 14 Independence When X and Y are discrete, X and Y are independent if and only if p ( x , y ) = p X ( x ) p Y ( y ) , for all x , y When X and Y are continuous, X and Y are independent if and only if f ( x , y ) = f X ( x ) f Y ( y ) , for all x , y When two random variables are not independent, we say that they are dependent .
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STAT 430/510 Lecture 14 Example Suppose that n + m independent trials having a common probability of success p are performed. If X is the number of successes in the first n trials, and Y is the number of successes in the final m trials. Are X and Y independent? P ( X = x , Y = y ) = ± n x ² p x ( 1 - p ) n - x ± m y ² p y ( 1 - p ) m - y = P ( X = x ) P ( Y = y ) X and Y are independent.
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Lecturenotes14 - STAT 430/510 Lecture 14 STAT 430/510...

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