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.
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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