Some applications of Dirac's delta function in Statistics for more than ...

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AAM: Intern, J., Vol. 3, Issue 1 (June 2008) [Previously, Vol. 3, No. 1] 49 In the multidimensional case, if the transformation is linear, i.e., Y = AX where Y and X are m × 1 and n × 1 vectors respectively and A is an m × n matrix, then we can express g (y) , the density of Y , in vector notation in terms of f (x) , the density of X as follows where So, ∞ g ∫ ∫ ( y) = ... f ( x) δ ( y − Ax) dx , (2) −∞ ∞ −∞ δ y ) = δ ( y )... δ ( y ) . ( 1 m δ ( y − Ax) = δ ( y1 − a1 x)... δ ( ym − am x) , T 1 T m T T where a ,..., a are the rows of the matrix A . Now, in the one-dimensional set up, if a is a δ ( x) scalar, then δ (ax) is given by δ ( ax) = . Similarly, in the multidimensional set up, if | a | Y = AX as above and A is a nonsingular matrix so that m = n , then, we must have −1 δ ( x − A y) δ ( y − Ax) = . (3) | A | This is because of the following: since the transformation is nonsingular, we have 1 −1 g ( y) = f ( A y) | A | and therefore, from (2) ∞ A y x A y Ax x 1 − ( ) = ... f ( ) | | δ ( − ) d . f ∫ ∫ But we know that ∞ −∞ ∞ ∫ ∫ −∞ ∞ −∞ −1 −1 ... f ( x) δ ( x − A y) dx = f ( A y) . −∞ Therefore, (3) follows. Similarly, if Y = AX + b where A is a nonsingular matrix and b is a vector, then, we have
50 Santanu Chakraborty −1 δ ( x − A (y − b)) δ ( y − (AX + b)) = . | A | Using this, one can conclude that if X is multivariate normal with μ as the mean vector and ∑ as the covariance matrix, then, for a nonsingular transformation A and a constant vector b , the transformed vector Y = AX + b follows multivariate normal with Aμ + b as the mean vector T and A ∑ A as the variance-covariance matrix. 7. Transformation of Variables in the Discrete Case Transformation of variables can be applied to the discrete case as well. If X is a discrete random variable taking the values n a a , , 1 with probabilities n p p , , 1 and if ) (X g Y = is a transformed variable, then the corresponding probability mass function for Y is given by ∞ ∫ p( x) ( y − g( x)) dx = ∑ q( y) = δ p δ ( y − g( a )) . −∞ n 1 In the two-dimensional case, if p ( x, y) is the joint probability mass function for the two variables X and Y , then q ( z, w) , the joint probability mass function for the transformed pair Z = φ ( X , Y ) and W = φ ( X , Y ) is given by 1 2 ∞ ∞ ( z, w) ∫ ∫ p( x, y) δ ( z −φ1 ( x, y)) δ ( w −φ 2 −∞−∞ m n q = ( x, y)) dxdy = ∑∑ 1 1 p δ ( z −φ ( a , b )) δ ( w −φ ( a , b )) ij 1 i j Where the pair ( X , Y ) is discrete having the values ( a i , b j ) with probabilities p ij for i = 1, 2,..., n and j = 1, 2,..., m . 8. Moments and Moment Generating Functions In the univariate set up, the ∞ k k ∫ x p x) dx = ∫ x ∑ −∞ ∞ −∞ th k non-central moment of X is written as 1≤i≤ n ∑ 2 1≤i≤ n i i ∞ j i k ∫ x ( x − ai ) dx = ∑ ( p δ ( x − a ) dx = p δ p a . i i i −∞ 1≤i≤ n In the bivariate set up, the non-central moment of order (k,l) for (X,Y) is given by ∞ ∞ ∫ ∫ −∞−∞ x k ∞ ∞ k l ∫ ∫ x y ∑∑ l y p( x, y) dxdy = p δ ( x − a ) δ ( y − b ) dxdy −∞−∞ m n 1 1 ij i j i k i
Page 1 and 2: Abstract Available at http://pvamu.
Page 3 and 4: 44 Santanu Chakraborty H ( x) = 0 f
Page 5 and 6: 46 Santanu Chakraborty We can also
Page 7: 48 Santanu Chakraborty the innermos
Page 11 and 12: 52 Santanu Chakraborty Then, ∞
Page 13: 54 Santanu Chakraborty = = ∑ ∑

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