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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: elementwise summation • Product: for X n × p = ( x ij ); B p × m = ( β jk ), their product Y = XB = ( y ik ) is a n by m matrix with y ik = ∑ p j =1 x ij β 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 di ff erent. 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( Ω ) = 1; Pr( Φ ) = 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...
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 Fall '09
 Normal Distribution, Probability theory, probability density function, Maximum likelihood, mle

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