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3. Multivariate Random Variables 131
Refer to (1.7.13), (1.7.20) and (1.7.31) as needed. Note that the
Theorem 3.5.3 really makes it simple to write down the marginal pdfs in
the case of independence. !
It is crucial to note that the supports χ s are assumed to
i
be unrelated to each other so that the representation given
by (3.5.6) may lead to the conclusion of independence among
the random variables X s. Look at the Example 3.5.11
i
to see what may happen when the supports are related.
Example 3.5.11 (Examples 3.3.1 and 3.5.2 Continued) Consider two
random variables X and X whose joint continuous distribution is given by
1 2
the following pdf:
Obviously, one has χ = χ = (0, 1). One may be tempted to denote,
1
2
for example, h (x ) = 6, h (x ) = (1 x ). At this point, one may be
1 1 2 2 2
tempted to claim that X and X are independent. But, that will be
1 2
wrong! On the whole space χ × χ , one can not claim that f(x , x )
2
2
1
1
= h (x )h (x ). One may check this out by simply taking, for ex-
1 1 2 2
ample, x = 1/2, x = 1/4 and then one has f(x , x ) = 0 whereas
1 2 1 2
h (x )h (x ) = 9/2. But, of course the relationship f(x , x ) =
2
1
2
1
1
2
h (x )h (x ) holds in the subspace where 0 < x < x < 1. From the
1 1 2 2 1 2
Example 3.5.2 one will recall that we had verified that in fact the
two random variables X and X were dependent. !
1 2
3.6 The Bivariate Normal Distribution
Let (X , X ) be a two-dimensional continuous random variable with
1
2
the following joint pdf:
with
The pdf given by (3.6.1) is known as the bivariate normal or two-dimensional
normal density. Here, µ , µ , σ , σ and ρ are referred to as the parameters
1 2 1 2