This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let's give it a try !!
Let and
be i.i.d. exponential random variables with mean
.Let
and
where
is a Bernoulli random variable with parameter
and is independent of
and
(a) Show that and
have the same distribution.
(b) Obtain the common density function.
Cumulative Distribution Function
Bernoulli distribution
Exponential Distribution
Cumulative distribution of be
,
Now,
Now, if then,
=
=
=
=
Now, then,
Therefore,
Cumulative distribution of be
,
=
=
=
since cdf of exponential random Variable, X is
Thus both and
has same distribution
(b)
=
Similarly, for .
If then find the distribution of
, where
.
This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let's give it a try !!
Let and
be i.i.d. exponential random variables with mean
.Let
and
where
is a Bernoulli random variable with parameter
and is independent of
and
(a) Show that and
have the same distribution.
(b) Obtain the common density function.
Cumulative Distribution Function
Bernoulli distribution
Exponential Distribution
Cumulative distribution of be
,
Now,
Now, if then,
=
=
=
=
Now, then,
Therefore,
Cumulative distribution of be
,
=
=
=
since cdf of exponential random Variable, X is
Thus both and
has same distribution
(b)
=
Similarly, for .
If then find the distribution of
, where
.