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# CLT and Confidence Limits | ISI MStat 2016 PSB Problem 8 This is a problem from ISI MStat Examination 2016. This primarily tests the student's knowledge in finding confidence intervals and using the Central Limit Theorem as a useful approximation tool.

## The Problem:

Let be independent and identically distributed pairs of random variables with , , and (a) Show that there exists a function such that : , where is the cdf of (b)Given , obtain a statistic which is a function of such that .

## Prerequisites:

(a)The Central Limit Theorem and Convergence of Sequence of RVs

(b)Idea of pivots and how to obtain confidence intervals.

## Solution:

(a) See that .

See that Take .

By the Central Limit Theorem,

see that as (b) Let  's are iid with and Again, by CLT, Use this as the pivot to obtain an asymptotic confidence interval for .

See that , where : upper point of .

Equivalently , you can write, , as .

Thus , .

## Food For Thought:

Suppose are equi-correlated with correlation coefficient .

Given, , , for .

Can you see that ?

It's pretty easy right? Yeah, I know 😛 . I am definitely looking forward to post inequalities more often .

This may look trivial but it is a very important result which can be used in problems where you need a non-trivial lower bound for .

Well,well suppose now .

Can you show that a necessary and sufficient condition for , to have equal variance is that should be uncorrelated with ?

Till then Bye!

Stay Safe.

This is a problem from ISI MStat Examination 2016. This primarily tests the student's knowledge in finding confidence intervals and using the Central Limit Theorem as a useful approximation tool.

## The Problem:

Let be independent and identically distributed pairs of random variables with , , and (a) Show that there exists a function such that : , where is the cdf of (b)Given , obtain a statistic which is a function of such that .

## Prerequisites:

(a)The Central Limit Theorem and Convergence of Sequence of RVs

(b)Idea of pivots and how to obtain confidence intervals.

## Solution:

(a) See that .

See that Take .

By the Central Limit Theorem,

see that as (b) Let  's are iid with and Again, by CLT, Use this as the pivot to obtain an asymptotic confidence interval for .

See that , where : upper point of .

Equivalently , you can write, , as .

Thus , .

## Food For Thought:

Suppose are equi-correlated with correlation coefficient .

Given, , , for .

Can you see that ?

It's pretty easy right? Yeah, I know 😛 . I am definitely looking forward to post inequalities more often .

This may look trivial but it is a very important result which can be used in problems where you need a non-trivial lower bound for .

Well,well suppose now .

Can you show that a necessary and sufficient condition for , to have equal variance is that should be uncorrelated with ?

Till then Bye!

Stay Safe.

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