Transcription of Properties of the Normal and Multivariate Normal …
1 Properties of the Normal and Multivariate Normal DistributionsBy Students of the Course, edited by Will WelchSeptember 28, 2014 Normal and Gaussian may be used Univariate Normal (Gaussian) DistributionLetYbe a random variable with mean (expectation) and variance 2> also Normal , and itsdistribution is denoted byN( , 2).In the followingaandbdenote constants, , they are not random Normal distributionN( , 2) has densityfY(y| , 2) =1 2 exp( 12 2(y )2)( < y < ). distribution is unimodal and the mode equals the mean equals the distribution belongs to the exponential generating ( , 2) distribution has MGFM(t) = exp( t+12 2t2),wheretis ( , 2)distribution has CF (t) =exp(i t 12 2t2), transformation.
2 (a) The distribution ofa+bYisN(a+ ,b2 2).(b) The distribution of (Y )/ isN(0,1), the standard density is infinitely Multivariate Normal (Gaussian) DistributionWe have a vector ofnrandom variables,Y= (Y1,..,Yn)T. Denote the mean (expectation) ofYiby i,and let = ( 1,.., n)Tbe then 1 vector of means. Similarly, is then nmatrix of , the random variables inYhave a joint Multivariate Normal distribution, denoted byMN( , ).We will assume the distribution is not degenerate, , is full rank, invertible, and hence positive vectoradenotes a vector of constants, , not random variables, in the following. Similarly,Bisa matrix of Multivariate Normal distributionMN( , ) has joint densityfY(y| , ) =1(2 )n/21det1/2( )exp( 12(y )T 1(y ))(y Rn).
3 1 Acadia Math 4013A1/5843A1, SFU STAT 890-4, UBC STAT 547 LProperties of the contours of the joint distribution aren-dimensional and covariance specify the ( , )joint distribution is specifiedby and generating ( , )distribution has MGFM(t) =exp( Tt+12tT t),wheretis a realn 1 ( , )distribution has CF (t) =exp(i Tt 12tT t), wheretis a realn 1 combinations.(a)Letaben ( , )if and only if any linear combinationaTYhas a (univariate) Normal distribution.(b) Letaben 1. The distribution ofaTYisN(aT ,aT a).(c) Letabem 1 andBbem n. The distribution of themrandom variablesa+BYisMN(a+B ,B BT).(d)LetZbenindependent standard Normal random variables. ThenY= +LZ, withLLT= ,has aMN( , ) distribution.(e) Again letLLT=.
4 ThenZ=L 1(Y ) has aMN(0,I) (a)YiandYjare independent if and only if ij= 0.(b) Pairwise independence ofYiandYjfor alli6=jimplies complete Marginal and Conditional Multivariate Normal (Gaussian) DistributionsLet then 1 vectorYwith aMN( , )distribution be partitioned asY= (Y1,Y2), whereY1andY2arem 1 and (n m) 1, respectively. Similarly, partition = ( 1, 2) and =( 11 12 21 22). marginal distribution ofY1isMN( 1, 11). distribution ofY1conditional onY2isMN( 1+ 12 122(Y2 2), 11 12 122 21). Y1andY2are independent if and only if 12= December 20142
