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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 !!

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