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This is a beautiful problem from ISI MStat 2016 Problem 5 (sample) PSB based on order statistics. We provide a detailed solution with the prerequisites mentioned explicitly.

Let and be independent and identically distributed Poisson random variables for some Let denote the corresponding order statistics.

(a) Show that

(b) Evaluate the limit of as the sample size .

- Pmf of k-th Order Statistic i.e
- Binomial Expansion
- Basic Inequality
- squeeze (or sandwich) theorem
- Limit

(a) Given , and be independent and identically distributed Poisson random variables for some Let denote the corresponding order statistics.

Let , F(j) be the CDF of i.e CDF of Poisson

Then , Pmf of k-th Order Statistic i.e

, where i.e the CDF of k-th Order Statistic

So,

Here we have to find ,

since , Poisson random variable takes values 0 ,1,2,.... i.e it takes all values < 0 with probabiliy 0 , that's why here for j=0 .

And , , as X follows Poisson .

So,

Therefore ,

.

Since , for and which is true hence our inequality hold's true (proved)

Hence , (proved )

(b) ( Using inequality in (a) )

So, -----(1)

As for i.e it's a fraction so it can be written as for some , Hence (Proof -Use l'hospital rule or think intutively that as n tends to infinity the exponential functions grows more rapidly than any polynomial function ).

Now taking limit in (1) , we get by **squeeze** (or **sandwich**) **theorem**

This is a beautiful problem from ISI MStat 2016 Problem 5 (sample) PSB based on order statistics. We provide a detailed solution with the prerequisites mentioned explicitly.

Let and be independent and identically distributed Poisson random variables for some Let denote the corresponding order statistics.

(a) Show that

(b) Evaluate the limit of as the sample size .

- Pmf of k-th Order Statistic i.e
- Binomial Expansion
- Basic Inequality
- squeeze (or sandwich) theorem
- Limit

(a) Given , and be independent and identically distributed Poisson random variables for some Let denote the corresponding order statistics.

Let , F(j) be the CDF of i.e CDF of Poisson

Then , Pmf of k-th Order Statistic i.e

, where i.e the CDF of k-th Order Statistic

So,

Here we have to find ,

since , Poisson random variable takes values 0 ,1,2,.... i.e it takes all values < 0 with probabiliy 0 , that's why here for j=0 .

And , , as X follows Poisson .

So,

Therefore ,

.

Since , for and which is true hence our inequality hold's true (proved)

Hence , (proved )

(b) ( Using inequality in (a) )

So, -----(1)

As for i.e it's a fraction so it can be written as for some , Hence (Proof -Use l'hospital rule or think intutively that as n tends to infinity the exponential functions grows more rapidly than any polynomial function ).

Now taking limit in (1) , we get by **squeeze** (or **sandwich**) **theorem**

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