Order Statistics of Poisson Distribution

by Sanket Agrawal   Last Updated April 15, 2019 11:19 AM - source

I have been given the following question,

Let $n ≥ 2$, and $X_1, X_2, . . . ,X_n$ be independent and identically distributed $Poisson (λ)$ random variables for some $λ > 0$. Let $X_{(1)} ≤ X_{(2)} ≤ · · · ≤ X_{(n)}$ denote the corresponding order statistics.

(a) Show that $P(X_{(2)} = 0) ≥ 1 − n(1 − e^{−λ})^{n−1}$.

(b) Evaluate the limit of $P(X_{(2)} > 0)$ as the sample size $n → ∞$.

I tried solving the question on my own and I have also been able to obtain the following expression; $P(X_{(2)}=0) = 1 - (1-e^{-\lambda})^n-ne^{-\lambda}(1-e^{-\lambda})^{n-1}$

$= 1-(1-e^{-\lambda})^{n-1}(1+e^{-\lambda}(n-1))$ ;

and it can be then shown that

$(1+e^{-\lambda}(n-1)) \le n \quad \text{for all } \lambda > 0 \text{ and } n \ge 2$ and thus

$P(X_{(2)} = 0) ≥ 1 − n(1 − e^{−λ})^{n−1}$

but I want to ask that is there any meaning of this statement. I mean is there any significance of the quantity on the left hand side of the equation so that the inequality can be derived intuitively or by any other method?



Related Questions


Using Chebyshev's inequality to obtain lower bounds

Updated January 24, 2018 17:19 PM



Poisson Distribution Question

Updated March 31, 2019 03:19 AM