lect02_03_090902_07

lect02_03_090902_07 - Some Basic Results in Probability...

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Unformatted text preview: Some Basic Results in Probability & Statistics Linear Algebra Probability Random Variables Common Statistical Distributions Statistical Estimation Statistical Inference about Normal Disbributions 2 Linear Algebra Summation and Product Operators n i =1 x i = x 1 + x 2 + + x n ; n i =1 Y i = Y 1 Y 2 Y n n i =1 p j =1 x ij = n i =1 { x i 1 + x ip } = x 11 + x 1 p + + x n 1 + x np Matrix: a rectangular display and organization of data. You can treat matrix as data with two subscripts, e.g. x ij , the first subscript is row index and the second is the column index. We note the matrix as X n p = ( x ij ), and call it a n by p matrix. 3 Matrix Operations Transpose: reverse the row and column index. So t ( X ) ij = x ji . Summation: element-wise summation Product: for X n p = ( x ij ); B p m = ( jk ), their product Y = XB = ( y ik ) is a n by m matrix with y ik = p j =1 x ij jk . Identity matrix I : square ( n = p ), diagonal equal to 1 and 0 elsewhere. Inverse: the product of a matrix X and its inverse X- 1 is identity matrix. Trace: for square matrix X n n , tr ( X ) = n i =1 x ii . 4 Some Notes about Matrix When doing matrix product XB , always make sure the number of columns of X and rows of B are equal. Matrix product has orders, XB and BX are different. For in- verse matrix we have XX- 1 = X- 1 X = I . So only square matrix has inverse. Only square matrix has trace, and tr ( XB ) = tr ( BX ). If X- 1 = t ( X ), we call X an orthogonal matrix. 5 Probability Sample space, events (sets) A,B Basic rules Pr() = 1; Pr() = 0 Pr( A B ) = Pr( A ) + Pr( B )- Pr( A B ) Pr( A B ) = Pr( A ) Pr ( B | A ) = Pr( B ) Pr( A | B ) Complementary events: Pr( A ) = 1- Pr( A ) 6 Random Variables...
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This note was uploaded on 12/06/2011 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.

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lect02_03_090902_07 - Some Basic Results in Probability...

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