More on Random Variables
Expectation :: Properties :: Pairs :: Independence :: Two R.V.S. :: Functions :: Inequalities :: Conditional :: General
Pairs of random variables
Ultimately, we will be dealing with infinite sequences of random variables. As steps along the way, we will examine carefully pairs of random variables, then vectors of random variables.
On
, the smallest
field of interest is
, which
is the smallest
field containing all of the rectangles. This
is the Borel
field of
.
That is,
Note that two r.v.s on form a bivariate r.v.
Properties of the joint c.d.f.:
 .
 .

, the marginal
c.d.f. of
.
, the marginal c.d.f. of .
 is continuous ''from the northeast.''
 is montonically increasing (or, more precisely, nondecreasing) in both variables.
Joint discrete r.v.s
Properties of
:
 , and if or
 .

Marginals:
Joint continuous r.v.s
Properties of joint p.d.f.:
 .
 .

We can get the p.d.f. from the c.d.f:

Marginals: