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?

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