30.12 PROPERTIES OF JOINT DISTRIBUTIONS
30.11.3 Marginal and conditional distributionsGiven a bivariate distributionf(x, y), we may be interested only in the proba-
bility function forXirrespective of the value ofY(or vice versa). Thismarginal
distribution ofXis obtained by summing or integrating, as appropriate, the
joint probability distribution over all allowed values ofY. Thus, the marginal
distribution ofX(for example) is given by
fX(x)={∑∫jf(x, yj) for a discrete distribution,
f(x, y)dy for a continuous distribution.(30.130)It is clear that an analogous definition exists for the marginal distribution ofY.
Alternatively, one might be interested in the probability function ofXgiventhatYtakes some specific value ofY=y 0 ,i.e.Pr(X=x|Y=y 0 ). Thisconditional
distribution ofXis given by
g(x)=f(x, y 0 )
fY(y 0 ),wherefY(y) is the marginal distribution ofY. The division byfY(y 0 ) is necessary
in order thatg(x) is properly normalised.
30.12 Properties of joint distributionsThe probability density functionf(x, y) contains all the information on the joint
probability distribution of two random variablesXandY. In a similar manner
to that presented for univariate distributions, however, it is conventional to
characterisef(x, y) by certain of its properties, which we now discuss. Once
again, most of these properties are based on the concept of expectation values,
which are defined for joint distributions in an analogous way to those for single-
variable distributions (30.46). Thus, the expectation value of any functiong(X, Y)
of the random variablesXandYis given by
E[g(X, Y)] ={∑
i∑
∫ jg(xi,yj)f(xi,yj) for the discrete case,
∞
−∞∫∞
−∞g(x, y)f(x, y)dx dy for the continuous case.30.12.1 MeansThe means ofXandYare defined respectively as the expectation values of the
variablesXandY. Thus, the mean ofXis given by
E[X]=μX={∑
i∑
jxif(xi,yj) for the discrete case,
∫∞
−∞∫∞
−∞xf(x, y)dx dy for the continuous case. (30.131)E[Y] is obtained in a similar manner.