This is a very simple and beautiful sample problem from ISI MStat PSB 2013 Problem 7. It is mainly based on simple hypothesis testing of normal variables where it is just modified with a bernoulli random variable. Try it!
Suppose and
are two independent and identically distributed random variables with
. Further consider a Bernoulli random variable
with
which is independent of
and
. Define
as,
For testing against
consider the test:
Rejects if
.
Find such that the test has size
.
Normal Distribution
Simple Hypothesis Testing
Bernoulli Trials
These problem is simple enough, the only trick is that to observe that the test rule is based on 3 random variables, and
but
on extension is dependent on the the other bernoulli variable
.
So, here it is given that we reject at size
if
such that,
So, Using law of Total Probability as, is conditioned on
,
[ remember,
, and
are independent of
].
Now, under ,
and
,
So, the rest part is quite obvious and easy to figure it out which I leave it is an exercise itself !!
Lets end this discussion with some exponential,
Suppose, are a random sample from
and
is another random sample from the population of
. Now you are to test
against
.
Can you show that the test can be based on a statistic such that,
.
What distribution you think, T should follow under null hypothesis ? Think it over !!
This is a very simple and beautiful sample problem from ISI MStat PSB 2013 Problem 7. It is mainly based on simple hypothesis testing of normal variables where it is just modified with a bernoulli random variable. Try it!
Suppose and
are two independent and identically distributed random variables with
. Further consider a Bernoulli random variable
with
which is independent of
and
. Define
as,
For testing against
consider the test:
Rejects if
.
Find such that the test has size
.
Normal Distribution
Simple Hypothesis Testing
Bernoulli Trials
These problem is simple enough, the only trick is that to observe that the test rule is based on 3 random variables, and
but
on extension is dependent on the the other bernoulli variable
.
So, here it is given that we reject at size
if
such that,
So, Using law of Total Probability as, is conditioned on
,
[ remember,
, and
are independent of
].
Now, under ,
and
,
So, the rest part is quite obvious and easy to figure it out which I leave it is an exercise itself !!
Lets end this discussion with some exponential,
Suppose, are a random sample from
and
is another random sample from the population of
. Now you are to test
against
.
Can you show that the test can be based on a statistic such that,
.
What distribution you think, T should follow under null hypothesis ? Think it over !!