2 EXPONENTIAL AND GAMMA DISTRIBUTIONS
Lecture 15
ORIE3500/5500 Summer2009 Chen
Class Today
•
Multivariate Normal (Gaussian) Distribution
•
Exponential and Gamma Distributions
1 Multivariate Normal Distribution
The bivariate normal distribution is one of the most important joint distrib
utions. (
X,Y
) are said to be a bivariate normal vector is they have density,
for all
,
∞
< x,y <
∞
,
f
X,Y
(
x,y
) =
1
2
πσ
1
σ
2
p
1

ρ
2
exp
‡

(
x

μ
1
)
2
σ
2
1

2
ρ
(
x

μ
1
)(
y

μ
2
)
σ
1
σ
2
+
(
y

μ
2
)
2
σ
2
2
2(1

ρ
2
)
·
.
The parameters
μ
1
,μ
2
,σ
1
,σ
2
and
ρ
, characterize the distribution.
μ
1
and
μ
2
are the means of
X
and
Y
respectively and
σ
2
1
,
σ
2
2
are the variance of
X
and
Y
respectively.
ρ
is the correlation between
X
and
Y
. We know that in
general if two variables are independent then they have 0 correlation. But
the other way is not true. That is , if two random variables are uncorrelated
then they are not necessarily independent. But if (
X,Y
) have bivariate joint
normal distribution and they are uncorrelated, then they are independent.
This is a property of the normal distribution.
The multivariate normal distribution is characterized by its mean and
variancecovariance matrix.
2 Exponential and Gamma distributions
Although the normal distribution ﬁnds widespread application in many areas,
it can not be used to model everything. For example, the waiting time of a
person in a queue, the service time of a customer in a counter, duration of
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 Summer '08
 WEBER
 Normal Distribution, Probability theory, Xi Gamma

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