Μ n x n μ n x 1 μ 1 x n μ n x 2 μ 2 x n μ n 2

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μ n ) . . . . . . . . . ( x n μ n )( x 1 μ 1 ) ( x n μ n )( x 2 μ 2 ) · · · ( x n μ n ) 2 . On applying the expectation operator to each of the elements, we get the matrix of variances and covariances. Quadratic products of the dispersion matrix. The summation vector ι = [1 , 1 , . . . , 1] is just a column of units. The inner product of the summation vector with a vector x = [ x 1 , x 2 , . . . , x n ] of the same order is just the sum of the elements of the latter vector: ι x = [1 1 . . . 1] x 1 x 2 . . . x n = x 1 + x 2 + · · · + x n = i x i . The variance of a the sum of the elements of a random vector x is given by V ( ι x ) = ι D ( x ) ι. To represent this more explicitly, we may write V ( ι x ) = [1 1 . . . 1] σ 11 σ 12 . . . σ 1 n σ 21 σ 22 . . . σ 2 n . . . . . . . . . σ n 1 σ n 2 . . . σ nn 1 1 . . . 1 . There are a variety of ways in which the quadratic products may be represented in scalar notation: V ( ι x ) = i j σ ij = i σ ii + i j σ ij i = j = i σ ii + 2 i j σ ij i < j. 2
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To generalise this result, let x = [ x 1 , x 2 , . . . , x n ] be a vector of random vari- ables and a = [ a 1 , a 2 , . . . , a n ] be a vector of constants. Then, the variance of the weighted sum a x = i a i x i is given V ( a x ) = a D ( x ) a . To see this, consider a D ( x ) a = a E x E ( x ) x E ( x ) = E a x E ( a x ) ax E ( ax ) = E a x E ( a x ) 2 = V ( a x ) .
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  • Spring '12
  • D.S.G.Pollock
  • Variance, Probability theory, Xn

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