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Let $X_1,X_2,...,X_n$ is a random sample of Poisson distribution with parameter one. if $T=\bar{X}(n-\bar{X})$ how can find upper bound for $P(T=0)$

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  • $\begingroup$ Presumably $\bar X$ the average? $\endgroup$ Feb 22, 2015 at 15:29

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We have $T=0$ iff all $X_i=0$ or $\sum_i X_i = n$. The sum of Poisson random variables is itself Poisson; in particular, $\sum_i X_i$ has Poisson($n$) distribution. Moreover, the aforementioned events are disjoint, so

$$ P(T=0) = P(X_1=0)^n + P(\sum X_i = n^2) = e^{-n} + e^{-n} n^{n^2} / n^2 ! $$

So you can write down the probability exactly.

Somehow I'm not sure if you really wanted $T$ defined like that though ?

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