note 9 - MA2216/ST2131 Probability Notes 9 Review...

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Unformatted text preview: MA2216/ST2131 Probability Notes 9 Review & Examples and Properties of Expectation § 1. Linear Transformation. Let X 1 ,X 2 ,...,X n be continuous random variables having joint density f and let random variables Y 1 ,Y 2 ,...,Y n be defined by the following linear transformation Y i = n X j =1 a ij X j , i = 1 , 2 ,...,n, where the matrix A = ( a ij ) n × n has nonzero determinant det A . Thus, ( y 1 ,...,y n ) t = A ( x 1 ,...,x n ) t , where ( x 1 ,...,x n ) t denotes the transpose of the row vector ( x 1 ,...,x n ). Since det A 6 = 0, the linear transformation A is non-singular, and hence admits an inverse, A- 1 such that ( x 1 ,...,x n ) t = A- 1 ( y 1 ,...,y n ) t . (1 . 1) Equivalently, x ∼ = y ∼ ( A- 1 ) t , where x ∼ = ( x 1 ,...,x n ). Note also that the Jacobian of this transformation A is nothing but its determinant: J ( x 1 ,x 2 ,...,x n ) = det A. Therefore, Y 1 ,Y 2 ,...,Y n have joint density f Y 1 ,...,Y n given by f Y 1 ,...,Y n ( y 1 ,...,y n ) = 1 | det A | f ( x 1 ,...,x n ) = 1 | det A | f ( y ∼ ( A- 1 ) t ) . (1 . 2) 1 § 2. Cauchy-Schwarz Inequality. 1. For sequences { a 1 ,a 2 ,...,a n } and { b 1 ,b 2 ,...,b n } , the classical Cauchy- Schwartz inequality is stated as follows: " n X i =1 a i b i # 2 ≤ ˆ n X i =1 a 2 i ! · ˆ n X i =1 b 2 i ! . If a ∼ = ( a 1 ,a 2 ,...,a n ) and b ∼ = ( b 1 ,b 2 ,...,b n ) are regarded as vectors in IR n , the above inequality becomes | a ∼ · b ∼ | ≤ || a ∼ |||| b ∼ || , where “ · ” refers to the inner product, and || a ∼ || def. = ˆ n X i =1 a 2 i ! 1 / 2 refers to its Euclidian norm. 2. Remarks. (i) The inequality becomes an equality if and only if a ∼ and b ∼ are linearly dependent , i.e., there exists a real constant t such that t · a ∼ = b ∼ . (ii) Such an inequality can be generalized to infinite series, say, { a i : i = 1 , 2 ,... } , provided that ∞ X i =1 a 2 i < ∞ . In this case, the Cauchy-Schwartz inequality is given by " ∞ X i =1 a i b i # 2 ≤ ˆ ∞ X i =1 a 2 i ! · ˆ ∞ X i =1 b 2 i ! . 2 3. Cauchy-Schwarz Inequality for R.V’s. Let X 1 and X 2 be random variables having joint density f ( x,y ). Assume also that X i has finite second moment for each i = 1 , 2. Then, ( EE [ XY ]) 2 ≤ EE [ X 2 ] EE [ Y 2 ] . (2 . 1) Proof. Unless Y =- tX for some constant t , in which case this in- equality holds with equality, it follows that for all t < EE [( tX + Y ) 2 ] = EE [ X 2 ] t 2 + 2 EE [ XY ] t + EE [ Y 2 ] . Hence the quadratic equation (in t ) EE [ X 2 ] t 2 + 2 EE [ XY ] t + EE [ Y 2 ] = 0 does not have real roots, which implies that the discriminant must be negative: (2 EE [ XY ]) 2 < 4 EE [ X 2 ] EE [ Y 2 ] . Thus, (2.1) is established, which is known as the Cauchy-Schwartz in- equality....
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This note was uploaded on 03/19/2012 for the course SCIENCE ST2131 taught by Professor Forgot during the Fall '08 term at National University of Singapore.

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note 9 - MA2216/ST2131 Probability Notes 9 Review...

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