Chapter 4: Part b – The Multivariate Normal Distribution
We will be discussing
 The Multivariate Normal Distribution
 Other Distributions
(These topics are needed for Chapters 5)
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Slide 4b.1
Distributions
The Multivariate Normal Function
According to the multivariate density function, the probability that the random
vector x = [x1 x2 ··· xp]′ takes on a particular set of values is given by
Pr(x  x a ) 
1
(2)
p/2
| Σ|
1/ 2
exp(x a  μ) Σ 1 (x a  μ) / 2.
The analogous distribution function is given by
1
(2) |Σ |
p/2
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1/ 2
   exp (x  μ) Σ
x a ,1 x a , 2
x a ,p
 

1
(x  μ) / 2dxp dxp1 dx1
Slide 4b.2
Distributions
Bivariate Normal with Three Values of 
 = 0.0
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 = 0.4
 = 0.6
Slide 4b.3
Distributions
The 2 Distribution
The Chi Square Is a Sum of Squared Z scores:
n
zz   z i2 ~  2n .
i
Pr(2)
12
0.25
It approaches normality as df gets large:
32
0.20
72
0.15
2
12
0.10
0.05
0.00
0
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5
10
15
20
25
Slide 4b.4
Distributions
Student’s t Distribution
The t is analogous to the normal but with 2 unknown. It
approaches normality also as the df gets large.
t 30
t1
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Slide 4b.5
Distributions
The F Distribution
The F is a ratio of Chi Squares. The t is an F2 with 1 df in the numerator.
Fr ,r 
1
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2
 2r r1
1

2
r2
r2
Slide 4b.6
Distributions