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Random vectors

1.7. Random vectors

Joint distributions

It is common in the technical sciences to consider simultaneously two or more random variables
that are not independent, in general, and are thus described by joint probability density
functions. We will return to the subject of the interdependence of random variables after first
presenting some of the general ways of characterizing joint distributions. We will concentrate
mainly on bivariate distributions, i.e. distributions of only two random variables, though the
results may be extended readily to multivariate distributions. The subject of multivariate
distributions is large and a detailed study is beyond the scope of this book; the interested reader
should therefore consult one of the many specialized texts. However, we do discuss the
multinomial and multivariate Gaussian distributions.
The first thing to note when dealing with bivariate distributions is that the distinction
between discrete and continuous distributions may not be as clear as for the single variable case;
the random variables can both be discrete, or both continuous, or one discrete and the other
continuous. In general, for the random variables X and Y, the joint distribution will take an
infinite number of values unless both X and Y have only a finite number of values. In this
chapter we will consider only the cases where X and Y are either both discrete or both
continuous random variables.

Discrete bivariate distributions. In direct analogy with the one-variable (univariate) case, if X
is a discrete random variable that takes the values {x i } and Y one that takes the values {y j } then
the probability function of the joint distribution is defined as

{
P(X = x i , Y = y j ) for x = x i , y = y j ,
f(x, y) =
0 otherwise .

We may therefore think of f(x, y) as a set of spikes at valid points in the xy-plane, whose heights
represent the probability of obtaining X = x i and Y = y j . The normalization of f(x, y) implies

∑ ∑
f(x i , y j ) = 1, (7.1)
i j

where the sums over i and j take all valid pairs of values. We can also define the cumulative
probability function (joint distribution function)

∑ ∑
F(x, y) = f(x i , y j ), (7.2)
x i ≤x y j ≤y

from which it follows that the probability that X lies in the range [a 1 , a 2 ] and Y lies in the range
[b 1 , b 2 ] is given by

P(a 1 < X ≤ a 2 , b 1 < Y ≤ b 2 ) = F(a 2 , b 2 ) − F(a 1 , b 2 ) − F(a 2 , b 1 ) + F(a 1 , b 1 ).

Finally, we define X and Y to be independent if we can write their joint distribution in the form

f(x, y) = f X (x)f Y (y), (7.3)

i.e. as the product of two univariate distributions.

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