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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 !!


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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 !!


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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