30.11 JOINT DISTRIBUTIONS
consult one of the many specialised texts. However, we do discuss the multinomial
and multivariate Gaussian distributions, in section 30.15.
The first thing to note when dealing with bivariate distributions is that thedistinction 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
variablesXandY, the joint distribution will take an infinite number of values
unless bothXandYhave only a finite number of values. In this chapter we
will consider only the cases whereXandY are either both discrete or both
continuous random variables.
30.11.1 Discrete bivariate distributionsIn direct analogy with the one-variable (univariate) case, ifXis a discrete random
variable that takes the values{xi}andYone that takes the values{yj}then the
probability function of the joint distribution is defined as
f(x, y)={
Pr(X=xi,Y=yj)forx=xi,y=yj,0otherwise.We may therefore think off(x, y) as a set of spikes at valid points in thexy-plane,
whose height at (xi,yi) represents the probability of obtainingX=xiandY=yj.
The normalisation off(x, y) implies
∑i∑jf(xi,yj)=1, (30.125)where the sums overiandjtake all valid pairs of values. We can also define the
cumulative probability function
F(x, y)=∑xi≤x∑yj≤yf(xi,yj), (30.126)from which it follows that the probability thatXlies in the range [a 1 ,a 2 ]andY
lies in the range [b 1 ,b 2 ] is given by
Pr(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 defineXandYto beindependentif we can write their joint distribution
in the form
f(x, y)=fX(x)fY(y), (30.127)i.e. as the product of two univariate distributions.