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This is a problem from the ISI MStat Entrance Examination, 2019. This primarily tests one's familiarity with size, power of a test and whether he/she is able to condition an event properly.

Let Z be a random variable with probability density function

with parameter . Suppose, we observe max .

(a)Find the constant c such that the test that "rejects when " has size 0.05 for the null hypothesis .

(b)Find the power of this test against the alternative hypothesis .

- A thorough knowledge about the size and power of a test
- Having a good sense of conditioning whenever a function (like max()) is defined piecewise.

And believe me as Joe Blitzstein says: **"Conditioning is the soul of statistics"**

(a) If you know what size of a test means, then you can easily write down the condition mentioned in part(a) in mathematical terms.

It simply means

Now, under , .

So, we have the pdf of Z as

As the support of Z is , we can partition it in .

Now, let's condition based on this partition. So, we have:

Do, you understand the last equality? (Try to convince yourself why)

So,

Equating with 0.05, we get

(b) The second part is just mere calculation given already you know the value of c.

Power of test against is given by:

The pdf occurring in this problem is an example of a **Laplace distribution**.Look it up on the internet if you are not aware and go through its properties.

Suppose you have a random variable V which follows Exponential Distribution with mean 1.

Let I be a Bernoulli() random variable. It is given that I,V are independent.

Can you find a function h (which is also a random variable), ( a continuous function of I and V) such that h has the standard Laplace distribution?

This is a problem from the ISI MStat Entrance Examination, 2019. This primarily tests one's familiarity with size, power of a test and whether he/she is able to condition an event properly.

Let Z be a random variable with probability density function

with parameter . Suppose, we observe max .

(a)Find the constant c such that the test that "rejects when " has size 0.05 for the null hypothesis .

(b)Find the power of this test against the alternative hypothesis .

- A thorough knowledge about the size and power of a test
- Having a good sense of conditioning whenever a function (like max()) is defined piecewise.

And believe me as Joe Blitzstein says: **"Conditioning is the soul of statistics"**

(a) If you know what size of a test means, then you can easily write down the condition mentioned in part(a) in mathematical terms.

It simply means

Now, under , .

So, we have the pdf of Z as

As the support of Z is , we can partition it in .

Now, let's condition based on this partition. So, we have:

Do, you understand the last equality? (Try to convince yourself why)

So,

Equating with 0.05, we get

(b) The second part is just mere calculation given already you know the value of c.

Power of test against is given by:

The pdf occurring in this problem is an example of a **Laplace distribution**.Look it up on the internet if you are not aware and go through its properties.

Suppose you have a random variable V which follows Exponential Distribution with mean 1.

Let I be a Bernoulli() random variable. It is given that I,V are independent.

Can you find a function h (which is also a random variable), ( a continuous function of I and V) such that h has the standard Laplace distribution?

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Why Cheenta?

Take Now since I can take or and so can take any real numbers as it's value. Let Then by using independence of the random variables and we get

Now if then again by using independence of the random variables and we get

Thus the random variable follows standard laplace distribution.

Great Work Arnab. Stay Tuned!