180 Decision Based Design
The individual random variables x
i
’s themselves follow distributions
represented by the cumulative density functions
i
. As we discussed
earlier, these single-variable distributions are referred to as marginal distri-
butions or marginals. For a function to qualify as a joint distribution function,
it has to satisfy the following four conditions:
1.
→−∞
lim (, ,..., )0forany {1,..., }
12X
x
n
i
2.
→∞
lim (, ,..., )1 {1,..., }
12X
Fxxx
x
n
i
3.
→∞
lim (,..., ,..., )()
1X
Fx xxFx
x
jn j
i
4.
⋅
X
F is n -increasing
The rst three conditions are pretty straightforward; condition 4, however,
needs a little explanation. An n -increasing function is the n -dimensional
version of an increasing function in one dimension. ...