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# ISI MStat PSB 2013 Problem 7 | Bernoulli interferes Normally

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!

## Problem- ISI MStat PSB 2013 Problem 7

Suppose $$X_1$$ and $$X_2$$ are two independent and identically distributed random variables with $$N(\theta, 1)$$. Further consider a Bernoulli random variable $$V$$ with $$P(V=1)=\frac{1}{4}$$ which is independent of $$X_1$$ and $$X_2$$ . Define $$X_3$$ as,

$$X_3 = \begin{cases} X_1 & when & V=0 \\ X_2 & when & V=1 \end{cases}$$

For testing $$H_o: \theta= 0$$ against $$H_1=\theta=1$$ consider the test:

Rejects $$H_o$$ if $$\frac{(X_1+X_2+X_3)}{3} >c$$.

Find $$c$$ such that the test has size $$0.05$$.

### Prerequisites

Normal Distribution

Simple Hypothesis Testing

Bernoulli Trials

## Solution :

These problem is simple enough, the only trick is that to observe that the test rule is based on 3 random variables, $$X_1,X_2$$ and $$X_3$$ but $$X_3$$ on extension is dependent on the the other bernoulli variable $$V$$.

So, here it is given that we reject $$H_o$$ at size $$0.05$$ if $$\frac{(X_1+X_2+X_3)}{3}> c$$ such that,

$$P_{H_o}(\frac{X_1+X_2+X_3}{3}>c)=0.05$$

So, Using law of Total Probability as, $$X_3$$ is conditioned on $$V$$,

$$P_{H_o}(X_1+X_2+X_3>3c|V=0)P(V=0)+P_{H_o}(X_1+X_2+X_3>3c|V=1)P(V=1)=0.05$$

$$\Rightarrow P_{H_o}(2X_1+X_2>3c)\frac{3}{4}+P_{H_o}(X_1+2X_2>3c)\frac{1}{4}=0.05$$ [ remember, $$X_1$$, and $$X_2$$ are independent of $$V$$].

Now, under $$H_o$$ , $$2X_1+X_2 \sim N(0,5)$$and $$X_1+2X_2 \sim N(0,5)$$ ,

So, the rest part is quite obvious and easy to figure it out which I leave it is an exercise itself !!

## Food For Thought

Lets end this discussion with some exponential,

Suppose, $$X_1,X_2,....,X_n$$ are a random sample from $$exponential(\theta)$$ and $$Y_1,Y_2,.....,Y_m$$ is another random sample from the population of $$exponential(\mu)$$. Now you are to test $$H_o: \theta=\mu$$ against $$H_1: \theta \neq \mu$$ .

Can you show that the test can be based on a statistic $$T$$ such that, $$T= \frac{\sum X_i}{\sum X_i +\sum Y_i}$$.

What distribution you think, T should follow under null hypothesis ? Think it over !!