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# ISI MStat PSB 2014 Problem 9 | Hypothesis Testing This is a another beautiful sample problem from ISI MStat PSB 2014 Problem 9. It is based on testing simple hypothesis, but reveals and uses a very cute property of Geometric distribution, which I prefer calling sister to Loss of memory . Give it a try !

## Problem- ISI MStat PSB 2014 Problem 9

Let and be independent random variables, where Geo(p) refers to Geometric distribution whose p.m.f. f is given by, We are interested in testing the null hypothesis against the alternative . Intuitively it is clear that we should reject if is large, but unfortunately, we cannot compute the cut-off because the distribution of under depends on the unknown (common) value and .

(a) Let . Find the conditional distribution of when .

(b) Based on the result obtained in (a), derive a level 0.05 test for against when is large.

### Prerequisites

Geometric Distribution.

Negative binomial distribution.

Discrete Uniform distribution .

Conditional Distribution . .

Simple Hypothesis Testing.

## Solution :

Well, Part (a), is quite easy, but interesting and elegant, so I'm leaving it as an exercise, for you to have the fun. Hint: verify whether the required distribution is Discrete uniform or not ! If you are done, proceed .

Now, part (b), is further interesting, because here we will not use the conventional way of analyzing the distribution of and , whereas we will be concentrating ourselves on the conditional distribution of ! But why ?

The reason behind this adaptation of strategy is required, one of the reason is already given in the question itself, but the other reason is more interesting to observe , i.e. if you are done with (a), then by now you found that , the conditional distribution of is independent of any parameter ( i.e. ithe distribution of looses all the information about the parameter , when conditioned by Y=y , is a necessary condition), and the parameter independent conditional distribution is nothing but a Discrete Uniform {0,1,....,y}, where y is the sum of and .

so, under , the distribution of is independent of the both common parameter and . And clearly as stated in the problem itself, its intuitively understandable , large value of exhibits evidences against . Since large value of is realized, means the success doesn't come very often .i.e. is smaller.

So, there will be strong evidence against if , where , for some constant , where y is given the sum of .

So, for a level 0.05 test , the test will reject for large value of k , such that, So, we reject at level 0.05, when we observe , where it is given that =y . That's it!

## Food For Thought

Can you show that for this same and , considering , where n=0,1,.... What about the converse? Does it hold? Find out!

But avoid loosing memory, it's beauty is exclusively for Geometric ( and exponential) !!

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This is a another beautiful sample problem from ISI MStat PSB 2014 Problem 9. It is based on testing simple hypothesis, but reveals and uses a very cute property of Geometric distribution, which I prefer calling sister to Loss of memory . Give it a try !

## Problem- ISI MStat PSB 2014 Problem 9

Let and be independent random variables, where Geo(p) refers to Geometric distribution whose p.m.f. f is given by, We are interested in testing the null hypothesis against the alternative . Intuitively it is clear that we should reject if is large, but unfortunately, we cannot compute the cut-off because the distribution of under depends on the unknown (common) value and .

(a) Let . Find the conditional distribution of when .

(b) Based on the result obtained in (a), derive a level 0.05 test for against when is large.

### Prerequisites

Geometric Distribution.

Negative binomial distribution.

Discrete Uniform distribution .

Conditional Distribution . .

Simple Hypothesis Testing.

## Solution :

Well, Part (a), is quite easy, but interesting and elegant, so I'm leaving it as an exercise, for you to have the fun. Hint: verify whether the required distribution is Discrete uniform or not ! If you are done, proceed .

Now, part (b), is further interesting, because here we will not use the conventional way of analyzing the distribution of and , whereas we will be concentrating ourselves on the conditional distribution of ! But why ?

The reason behind this adaptation of strategy is required, one of the reason is already given in the question itself, but the other reason is more interesting to observe , i.e. if you are done with (a), then by now you found that , the conditional distribution of is independent of any parameter ( i.e. ithe distribution of looses all the information about the parameter , when conditioned by Y=y , is a necessary condition), and the parameter independent conditional distribution is nothing but a Discrete Uniform {0,1,....,y}, where y is the sum of and .

so, under , the distribution of is independent of the both common parameter and . And clearly as stated in the problem itself, its intuitively understandable , large value of exhibits evidences against . Since large value of is realized, means the success doesn't come very often .i.e. is smaller.

So, there will be strong evidence against if , where , for some constant , where y is given the sum of .

So, for a level 0.05 test , the test will reject for large value of k , such that, So, we reject at level 0.05, when we observe , where it is given that =y . That's it!

## Food For Thought

Can you show that for this same and , considering , where n=0,1,.... What about the converse? Does it hold? Find out!

But avoid loosing memory, it's beauty is exclusively for Geometric ( and exponential) !!

## Subscribe to Cheenta at Youtube

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