Lecture 4-Probability Slides

X32 2 dx 2 53 univariate normalgaussian density what

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Unformatted text preview: 3)2 √ e 2 dx = 2π −∞ 49 Univariate normal/Gaussian density What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ This is just the integral over a Gaussian pdf with mean µ 3 and variance σ 2 1 50 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −∞ ? −(x−3)2 dx 2 = 51 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −∞ −(x−3)2 dx 2 = √ 2π 52 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −(x−3)2 2 dx = √ −∞ What is 3 e −∞ ? −(x−3)2 2 dx = 2π 53 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −(x−3)2 2 e −(x−3)2 2 dx = √ 2π −∞ What is 3 −∞ √ dx = 2π 2 This is the integral under half of the pdf (from neg infinity to the mean of a symmetric density) 54 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −(x−3)2 2 e −(x−3)2 2 dx = √ 2π −∞ What is 3 −∞ What is ? ∞ √ dx = 2π 2 1 −(x−3)2 x √ e 2 dx = 2π −∞ 55 Univariate normal/Gaussian density What is What is ∞ 1 −(x−3)2 √ e 2 dx = 1 2π −∞ ∞ e −(x−3)2 2 e −(x−3)2 2 dx = √ 2π −∞ What is 3 −∞ What is √ dx = 2π 2 ∞ 1 −(x−3)2 x √ e 2 dx = 3 2π −∞ Note how this is xp(x)dx = E (X ) 56 Short aside on Determinants (of square matrices) We can define the determinant inductively The determinant of a 2 × 2 Matrix A= ab cd is |A| = ad − bc 57 Determinant cont’d The determinant of a n × n matrix A (only defined for square matrices) is det A = a11M11 − a21M21 + ... + (−1)n+1an1Mn1 where Mij is the determinant of the (n − 1) × (n − 1) matrix obtained by removing the ith row and jth column of A. 58 In matlab use det(A) >> A = [ 1 2 ; 3 4] A= 1 3 >> det(A) ans = -2 2 4 59 Abs(Determinant) gives the Volume The absolute value of the determinant gives the volume of a parallelepiped having as adjacent edges t he row vectors of the matrix r1 r2 r3 A = s 1 s2 s3 t1 t2 t3 The determinant is also the product of the eigenvalues and is in general a measure of the volume changing property of the mapping A. (The sign tells whether the transformation preserves orientation, the absolute value gives the scale factor of 60 61 the transformation)). det(A) = λi i Multivariate normal density p(x) = 1 n 2 (2π ) (det Σ) 1 2 e (x−µ)T Σ−1 (x−µ) − 2 Contours of constant density are defined by x such that (x − µ) Σ−1(x − µ) = c2 Ellipses are centered at µ with axes ± (λi)ei where λi and ei are the eigenvalues and eigenvectors of Σ (this will be relevant for PCA and related algorithms) • Linear combinations of the components of X are normally distributed • All subsets of the components of X are normally distributed • zero entries in the covariance matrix implies that the corresponding components are independent 62 • The conditional distributions of the components are multivariate normal 63 Multivariate Gaussians 64...
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