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
.
(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
.
(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