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Random Variables and Probability Distributions 63

Now we see that Equation (3.38) leads to
dF XY xjy† f XY x; y†
f XY xjy†ˆ ˆ ; f y† 6ˆ 0; 3:42†
Y
dx f y†
Y
which is in a form identical to that of Equation (3.35) for the mass functions – a
satisfying result. We should add here that this relationship between the condi-
tional density function and the joint density function is obtained at the expense
of Equation (3.33) for F XY (x y). We say ‘at the expense of’ because the defin-
j
j
ition given to F XY (x y) does not lead to a convenient relationship between
j
F XY (x y) and F XY (x, y), that is,
F XY x; y†
F XY xjy† 6ˆ : 3:43†
F Y y†
This inconvenience, however, is not a severe penalty as we deal with density
functions and mass functions more often.
When random variables X and Y are independent, F XY (x y) ˆ F X (x) and, as
j
seen from Equation (3.42),
f XY xjy†ˆ f x†; 3:44†
X
and
f x; y†ˆ f x†f y†; 3:45†
XY X Y
which shows again that the joint density function is equal to the product of the
associated marginal density functions when X and Y are independent.
Finally, let us note that, when random variables X and Y are discrete,

i : x i x
X
F XY xjy†ˆ p XY x i jy†; 3:46†
iˆ1
and, in the case of a continuous random variable,

Z x
F XY xjy†ˆ f ujy†du: 3:47†
XY
1
Comparison of these equations with Equations (3.7) and (3.12) reveals they are
identical to those relating these functions for X alone.
Extensions of the above results to the case of more than two random vari-
ables are again straightforward. Starting from

P ABC†ˆ P AjBC†P BjC†P C†

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