Fundamentals of Probability and Statistics for Engineers

N ow we see that Equation (3.38) leads to

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 FX Y (xy). We say ‘at the expen se of’ because the defin-
ition given to FX Y (xy) does not lead to a convenient relationship between
FX Y (xy) and FX Y (x, y), that is,

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, FX Y (xy) FX (x) and, as
seen from Equation (3.42),

and

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,

and, in the case of a continuous random variable,

Comparison of these equations with Equations (3.7) and (3.12) reveals they are
id entical 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

Random Variables and Probability D istributions 63

fXY…xjy†ˆ

dFXY…xjy†

dx

ˆ

fXY…x;y†

fY…y†

; fY…y†6ˆ 0 ; … 3 : 42 †

j

j

j

FXY…xjy†6ˆ

FXY…x;y†

FY…y†

: … 3 : 43 †

jˆ

fXY…xjy†ˆfX…x†; … 3 : 44 †

fXY…x;y†ˆfX…x†fY…y†; … 3 : 45 †

FXY…xjy†ˆ

i:Xxix

iˆ 1

pXY…xijy†; … 3 : 46 †

FXY…xjy†ˆ

Zx

1

fXY…ujy†du: … 3 : 47 †

P…ABC†ˆP…AjBC†P…BjC†P…C†