Bernoulli distribution - Exercise set 1
This exercise set contains some solved exercises on the Bernoulli distribution. The theory needed to solve these exercises is introduced in the lecture entitled Bernoulli distribution.
Exercise 1.1
Let
and
be two independent Bernoulli random variables with parameter
.
Derive the probability mass function of their
sum:
The probability mass function of
is:![[eq2]](http://images1.statlect.com/bernoulli_distribution_exercise_set_1__6.png)
The
probability mass function of
is:![[eq3]](http://images2.statlect.com/bernoulli_distribution_exercise_set_1__8.png)
The
support of
(the set of values
can take) is:
The
formula for the probability mass function of a sum
of two independent variables
is:![[eq5]](http://images1.statlect.com/bernoulli_distribution_exercise_set_1__12.png)
where
is the support of
.
When
,
the formula
gives:![[eq6]](http://images2.statlect.com/bernoulli_distribution_exercise_set_1__16.png)
When
,
the formula
gives:![[eq7]](http://images1.statlect.com/bernoulli_distribution_exercise_set_1__18.png)
When
,
the formula
gives:![[eq8]](http://images2.statlect.com/bernoulli_distribution_exercise_set_1__20.png)
Therefore,
the probability mass function of
is:![[eq9]](http://images1.statlect.com/bernoulli_distribution_exercise_set_1__22.png)
Exercise 1.2
Let
be a Bernoulli random variable with parameter
.
Find its tenth moment.
The moment
generating function of
is:The
tenth moment of
is equal to the tenth derivative of
its moment generating function, evaluated at
:![[eq11]](http://images2.statlect.com/bernoulli_distribution_exercise_set_1__29.png)
Butso
that:![[eq13]](http://images1.statlect.com/bernoulli_distribution_exercise_set_1__31.png)