 Get inspired by the success stories of our students in IIT JAM 2021. Learn More

# Size, Power, and Condition | ISI MStat 2019 PSB Problem 9

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.

## The Problem:

Let Z be a random variable with probability density function

$f(z)=\frac{1}{2} e^{-|z- \mu|} , z \in \mathbb{R}$ with parameter $\mu \in \mathbb{R}$. Suppose, we observe $X =$ max $(0,Z)$.

(a)Find the constant c such that the test that "rejects when $X>c$" has size 0.05 for the null hypothesis $H_0 : \mu=0$.

(b)Find the power of this test against the alternative hypothesis $H_1: \mu =2$.

## Prerequisites:

• 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"

## Solution:

(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 $P_{H_0}(X>c)=0.05$

Now, under $H_0$, $\mu=0$.

So, we have the pdf of Z as $f(z)=\frac{1}{2} e^{-|z|}$

As the support of Z is $\mathbb{R}$, we can partition it in $\{Z \ge 0,Z <0 \}$.

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

$P_{H_0}(X > c)=P_{H_0}(X>c , Z \ge 0)+ P_{H_0}(X>c, Z<0) =P_{H_0}(X>c , Z \ge 0) =P_{H_0}(Z > c)$

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

So, $P_{H_0}(X >c)=P_{H_0}(Z > c)=\int_{c}^{\infty} \frac{1}{2} e^{-|z|} dz = \frac{1}{2}e^{-c}$

Equating $\frac{1}{2}e^{-c}$ with 0.05, we get $c= \ln{10}$

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

Power of test against $H_1$ is given by:

$P_{H_1}(X>\ln{10})=P_{H_1}(Z > \ln{10})=\int_{\ln{10}}^{\infty} \frac{1}{2} e^{-|z-2|} dz = \frac{e^2}{20}$

## Try out this one:

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($\frac{1}{2}$) random variable. It is given that I,V are independent.

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

# Knowledge Partner  