Summary
Summary Maths Stats 246 Part 2
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These notes summarise the first part of the maths stats 246 syllabus at stellenbosch university. the notes include theory as well as numerous worked examples with explanation.
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DefinetheJacobian matrix of h x asthe pep matrixJacobian
oftransformation
and the Jacobian Jh of h x as the determinant of thematrix
If Jh to then the Pdf of Y follows as
fy ly fi h ly July yeB
Where
July is the Jacobian of the inversetransformation X K ly
willbetestedinAl
July Shily Shily hilly
dy bys Jyp
Shelly
dy i
Jhptly Ohp
Ily
dy Jyp
at
Example X and fx x e given a so as o
Transformation se and 2 04 22
y y
Now inverse transformation
dry
Ka
ya a yay
, dy
ily
Now Jh
JF dy
I 1
dlyzyidlyz.gl
y
diet ili ok 1 1
July
fyly fi Hlf Jhly elyityz.gl
fy ly EY a o x2 o
o
gro yay
fyly let o
ocyicys
elsewhere yay
, chapterTwo
Introduction
We will now studythe multivariate distribution which is an extensionofthe univariate
distribution The Pvariate distribution is a multivariate distribution with P Variables
The P Variate normal distribution is important for severalreasons the behaviour of
bythe p variate normal
world can be
many occurrences inthereal
modelled
distribution
Evenif thedata distributed
is not accordingtothemultivariate normal distribution
the mean vector or vector oftotals of a random sample canbe approximated
bythe P Variate normal distribution viathe central Limittheorem
Variate NormalDistribution
Forthe univariate case X has normal distribution if
x normally04 then
g eta É o Cocco
PDFnormaldistribution
Thiscanbe rearranged tothe form
2 X Blax B
fx x ke
where 2 and k are chosen sothattheintegraloverthe full rangeequalsone
, NowthePDFofthe PVariatenormal distribution
written
slightlydifferently
Forthe univariate case fx x angleayin é
Nowforthe multivariatecase we havemorethanone random variable wehave a random
vector containing prandomvariables
P
vectorfollows variatenomaldistribution
covariancematrix
nomalp ie g
xp
eachone is a
If
randomvariableand
where
have ajoint m
they
distribution
If Oc Pa hasthe p dimensionalnormal distribution withexpected value vector M and
covariance matrix E pxp ie X normalp M E thenthe Pdfof X pxl is
k X M E d u
f x Gc anyPiz e la e
verysimilarformto univariate
P
s f x Gc at E exp 12 X M E X m
Donotneed to knowhowtoderive
example of
Pdfforbivariate
case
plotwhen
Wecanonly
bivariate iewhenP2
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