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Lecture_12 - 1 LINEAR COMBINATIONS OF N RANDOM VARIABLES...

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1 LINEAR COMBINATIONS OF N RANDOM VARIABLES Lecture 12 ORIE3500/5500 Summer2009 Chen Class Today Covariance (cont.) Correlation Coefficient 1 Linear combinations of n random variables If X 1 , . . . , X n are random variables and a 1 , . . . , a n are constants, then a 1 X 1 + · · · + a n X n is a linear combination of the random variables. We already know that for X 1 , X 2 and Y cov ( X 1 + X 2 , Y ) = cov ( X 1 , Y ) + cov ( X 2 , Y ) . we can generalize this further, that is, for random variables X 1 , . . . , X n and Y , cov ( n X i =1 X i , Y ) = n X i =1 cov ( X i , Y ) . For X 1 , X 2 , Y 1 and Y 2 , we have cov ( X 1 + X 2 , Y 1 + Y 2 ) = cov ( X 1 , Y 1 + Y 2 ) + cov ( X 2 , Y 1 + Y 2 ) = cov ( X 1 , Y 1 ) + cov ( X 1 , Y 2 ) + cov ( X 2 , Y 1 ) + cov ( X 2 , Y 2 ) . Going a bit further for random variables X 1 , . . . , X n and Y 1 , . . . , Y m , cov ( n X i =1 X i , m X j =1 Y j ) = n X i =1 m X j =1 cov ( X i , Y j ) . Extending this even further, if a 1 , . . . , a n and b 1 , . . . , b m , then cov ( n X i =1 a i X i , m X j =1 b j Y j ) = n X i =1 m X j =1 a i b j cov ( X i , Y j ) . 1
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1 LINEAR COMBINATIONS OF N RANDOM VARIABLES Example It will become easy to see something we have already seen before using the linearity property of covariance. For any two random variables X and Y , var ( X + Y ) = cov ( X + Y, X + Y ) = cov ( X, X ) + cov ( X, Y ) + cov ( Y, X ) + cov ( Y, Y ) = var ( X ) + var ( Y ) + 2 cov ( X, Y ) . The implication of this fact is that var ( X + Y ) = var ( X ) + var ( Y ) if and only if cov ( X, Y ) = 0. Taking hint from the last example we can generalize what variance of linear combinations of random variable will be. For any random variables X 1 , . . . , X n var ( n X i =1 X i ) = n X i =1 var ( X i ) + 2 n X i =1 i - 1 X j =1 cov ( X i , X j ) . Similarly, if a 1 , . . . , a n are constants, then, var ( n X i =1 a i X i ) = n X i =1 a 2 i var ( X i ) + 2 n X i =1 i - 1 X j =1 a i a j cov ( X i , X j ) .
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