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Gaussian Probability Density Functions: Properties and ...
Gaussian Probability Density Functions: Properties and Error CharacterizationMaria Isabel RibeiroInstitute for Systems and RoboticsInstituto Superior TcnicoAv. Rovisco Pais, 11049-001 Lisboa M. Isabel Ribeiro, 2004 February 2004 Contents1 Normal random variables22 Normal random for second order .. of constant Probability .. 143 Properties224 Covariance matrices and error ellipsoid241 chapter 1 Normal random variablesA random variableXis said to be normally distributed with mean and variance 2if its Probability Density function (pdf) isfX(x) =1 2 exp[ (x )22 2], < x < .( )Whenever there is no possible confusion between the random variableXand thereal argument,x, of the pdf this is simply represented byf(x)omitting the explicitreference to the random variableXin the subscript. The Normal or Gaussiandistribution ofXis usually represented by,X N( , 2),or also,X N(x , 2).The Normal or Gaussian pdf ( ) is a bell-shaped curve that is symmetric aboutthe mean and that attains its maximum value of1 2 ' atx= asrepresented in Figure for = 2and 2= Gaussian pdfN( , 2)is completely characterized by the two parameters and 2, the first and second order moments, respectively, obtainable from thepdf as =E[X] = xf(x)dx,( ) 2=E[(X )2] = (x )2f(x)dx( )2 6 4 (x)Figure : Gaussian or Normal pdf, N(2, )The mean, or the expected value of the variable, is the centroid of the pdf.
Chapter 1 Normal random variables A random variable X is said to be normally distributed with mean µ and variance σ2 if its probability density function (pdf) is f X(x) = 1 √ 2πσ exp − (x−µ)2 2σ2 , −∞ < x < ∞. (1.1) Whenever there is no possible …
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