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3. Multivariate Random Variables 125

so that we have

It is not hard to see that E[Y Z ] = 0, E[Y ] = p , E[Z ] = p , and hence we can
i
j
1 1
1
1
rewrite (3.4.16) to claim that
for all fixed i ≠ j = 1, ..., k.
This negative covariance should intuitively make sense because out of the
n marbles, large number of marbles in box #i would necessarily force the
number of marbles in the box #j to be small. Next, one can simply obtain

for all fixed i ≠ j = 1, ..., k.

3.5 Independence of Random Variables
Suppose that we have a k-dimensional random variable X = (X , ..., X ) whose
1
k
joint pmf or pdf is written as f(x) or f(x , ..., x ) with x ∈ χ (⊆ ℜ), i = 1, ...,
1
i
i
k
k. Here χ is the support of the random variable X , i = 1, ..., k, where these
i
i
random variables can be discrete or continuous.
Definition 3.5.1 Let f (x ) denote the marginal pmf or pdf of X , i = 1, ...,
i
i
i
k. We say that X , ..., X form a collection of independent random variables if
1
k
and only if