Two (or more) random variables can be defined over the same
sample space. Just as with jointly defined events, the
joint distribution function is easily defined.
The
joint probability density function
is related to the distribution function via double
integration.
or
Since
, the so-called
marginal density functions can be related to the joint density function.
and
Extending the ideas of conditional probabilities, the
conditional probability density function
is defined (when
) as
Two random variables are
statistically independent when
, which is equivalent to the condition that the joint
density function is separable:
.
For jointly defined random variables, expected values are
defined similarly as with single random variables. Probably themost important joint moment is the
covariance :
where
Related to the covariance is the (confusingly named)
correlation coefficient : the covariance normalized
by the standard deviations of the component random variables.
When two random variables are
uncorrelated , their
covariance and correlation coefficient equals zero so that
. Statistically independent random variables are
always uncorrelated, but uncorrelated random variables can bedependent.
Let
be uniformly distributed over
and let
. The two random variables are uncorrelated, but are
clearly not independent.
A
conditional expected value is the mean of the
conditional density.
Note that the conditional expected value is now a function of
and is therefore a random
variable. Consequently, it too has an expected value, which is
easily evaluated to be the expected value of
.
More generally, the expected value of a function of two random
variables can be shown to be the expected value of a conditionalexpected value:
. This kind of calculation is frequently simpler to
evaluate than trying to find the expected value of
"all at once." A particularly interesting example of
this simplicity is the
random sum of random
variables . Let
be a
random variable and
a sequence of random variables. We will find occasion
to consider the quantity
. Assuming that each component of the sequence has the
same expected value
, the expected value of the sum is found to be
