# Order Statistics of Poisson Distribution

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

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?

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