Independent random variables - Exercise set 1
This exercise set contains some solved exercises on independent random variables. The theory needed to solve these exercises is introduced in the lecture entitled Independent random variables.
Exercise 1.1
Consider two random variables
and
having marginal distribution
functions![[eq1]](http://images1.statlect.com/independent_random_variables_exercise_set_1__3.png)
If
and
are independent, what is their joint distribution function?
For
and
to be independent, their joint distribution function must be equal to the
product of their marginal distribution
functions:![[eq2]](http://images2.statlect.com/independent_random_variables_exercise_set_1__8.png)
Exercise 1.2
Let
be a discrete random vector with support:
Let
its joint probability mass function
be:Are
and
independent?
In order to verify whether
and
are independent, we first need to derive the marginal probability mass
functions of
and
.
The support of
is:and
the support of
is:We
need to compute the probability of each element of the support of
:![[eq8]](http://images1.statlect.com/independent_random_variables_exercise_set_1__23.png)
Thus,
the probability mass function of
is:![[eq9]](http://images2.statlect.com/independent_random_variables_exercise_set_1__25.png)
We
need to compute the probability of each element of the support of
:![[eq10]](http://images1.statlect.com/independent_random_variables_exercise_set_1__27.png)
Thus,
the probability mass function of
is:![[eq11]](http://images2.statlect.com/independent_random_variables_exercise_set_1__29.png)
The
product of the marginal probability mass functions
is:![[eq12]](http://images1.statlect.com/independent_random_variables_exercise_set_1__30.png)
which
is equal to
.
Therefore,
and
are independent.
Exercise 1.3
Let
be an absolutely
continuous random vector with support:
and
its joint probability density function
be:![[eq16]](http://images2.statlect.com/independent_random_variables_exercise_set_1__36.png)
Are
and
independent?
The support of
is:When
,
the marginal
probability density function of
is
,
while, when
,
the marginal probability density function of
is:Thus,
summing up, the marginal probability density function of
is:The
support of
is:When
,
the marginal probability density function of
is
,
while, when
,
the marginal probability density function of
is:Thus,
the marginal probability density function of
is:Verifying
that
is straightforward. When
or
,
then
.
When
and
,
then:![[eq33]](http://images1.statlect.com/independent_random_variables_exercise_set_1__65.png)
Thus,
and
are independent.