Multivariate Normal Distribution
p. 3-7
• Density of Normal Distribution
¾ univariate case: normal density
¾ multivariate case:
(Note.
„
must exist, i.e., |Σ|≠0)
the term
is a distance measure
„
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
¾ Example: bivariate normal density
„
„
„
„
„
if X1 and X2 are uncorrelated, they are independent
p. 3-8
p. 3-9
„
contour for the case
‹
‹
‹
⇒
‹
‹
What if |ρ12|
increase?
‹
when
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
(a) σ1=σ2, ρ=0 ⇒ independent and equal variance
joint pdf
contour lines of the pdf
(b) σ1=σ2, ρ=0.75 ⇒ correlated and equal variance
p. 3-10
data generated from the pdf
Q: what should the contour
lines look like if σ1≠σ2?
when σ1=σ2, ρ≠0, the major/minor axis of the ellipse is parallel to x1=x2 or x1=−x2
contour of Normal pdf is an ellipse because it can be expressed as (x−μ)T∑−1(x−μ)=c
p. 3-11
¾ random sample from a multivariate normal distribution
¾ Q: why normal?
„
While real data are never exactly multivariate normal, the normal density is
often a useful approximation to the “true” population distribution
„
The multivariate normal density is mathematically tractable and “nice”
results can be obtained
„
The distribution of many multivariate statistics are approximately normal,
regardless of the form of the parent population because of a central limit
effect
• Some properties of multivariate normal distribution
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
¾
p. 3-12
Alternative definition of multivariate normal distribution: Let z1 , . . . , zp be i.i.d
from N (0, 1) and Z = [z1 , . . . , zp ]T . For a p-dim vector μ and a p × p symmetric,
positive definite matrix Σ, X is said to have a multivariate normal distribution
Np (μ, Σ) if it has the same distribution as
Σ1/2 Z + μ
„
p. 3-13
¾
¾
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
Note. Suppose that X1 and X2 are multivariate normal. X =
not be a multivariate normal.
¾
p. 3-14
X1
X2
may
p. 3-15
„
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
Example: conditional density of a bivariate normal distribution
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‹
¾
conditional density and regression model
p. 3-16