Covariance - Exercise set 2
This exercise set contains some solved exercises on the computation of the covariance between absolutely continuous random variables. The theory needed to solve these exercises is introduced in the lecture entitled Covariance.
Exercise 2.1
Let
![[eq1]](s.gif)
be an absolutely continuous random vector
with support:
i.e.
is the set of all couples
such that
and
.
Let the joint probability density function of
be:![[eq6]](http://images1.statlect.com/covariance_exercise_set_2__8.png)
Compute
the covariance between
and
.
The support of
is:thus,
when
,
the marginal probability density function of
is
,
while, when
,
the marginal probability density function of
is:Therefore,
the marginal probability density function of
is:![[eq11]](http://images2.statlect.com/covariance_exercise_set_2__20.png)
The
expected value of
is:![[eq12]](http://images1.statlect.com/covariance_exercise_set_2__22.png)
The
support of
is:When
,
the marginal probability density function of
is
,
while, when
,
the marginal probability density function of
is:Therefore,
the marginal probability density function of
is:The
expected value of
is:The
expected value of the product
can be computed using the transformation
theorem:![[eq19]](http://images2.statlect.com/covariance_exercise_set_2__36.png)
Hence,
using the covariance formula, the covariance between
and
is:
Exercise 2.2
Let
be an absolutely continuous random vector
with support:
and
its joint probability density function
be:![[eq23]](http://images1.statlect.com/covariance_exercise_set_2__42.png)
Compute
the covariance between
and
.
The support of
is:When
,
the marginal probability density function of
is
,
while, when
,
the marginal probability density function of
is:Putting
pieces together, the marginal probability density function of
is:The
expected value of
is:The
support of
is:When
,
the marginal probability density function of
is
,
while, when
,
the marginal probability density function of
is:We
do not explicitly compute the integral, but we write the marginal probability
density function of
as
follows:![[eq34]](http://images2.statlect.com/covariance_exercise_set_2__66.png)
The
expected value of
is:![[eq35]](http://images1.statlect.com/covariance_exercise_set_2__68.png)
The
expected value of the product
can be computed using the transformation
theorem:![[eq36]](http://images2.statlect.com/covariance_exercise_set_2__70.png)
Hence,
using the covariance formula, the covariance between
and
is:
Exercise 2.3
Let
and
be two random variables such
that:
Compute the following
covariance:
Using bilinearity of the covariance
operator:![[eq40]](http://images1.statlect.com/covariance_exercise_set_2__78.png)