Prove that the following sequences monotnically decrease and increase correspondingly. Since they are bounded, find the limit.

Let $\xi_n$ be a Poisson random variable with $\lambda = n \in \mathbb{N}$. That is $P(\xi_n = k) = \frac{n^ke^{-n}}{k!}$, for $k \in \mathbb{N}_0$. Let $f_+(n) = P(\xi_n \geq n),\ f_-(n) = P(\xi_n > n)$. Show that $f_+(n)$ monotnically decreases and $f_-(n)$ monotnically increases. Find the limit of these two sequences.

Our professor gave us this task. I have no idea how to prove this, I've tried mathematical induction and it didn't make any sense.

All what I've succeded in is that $f_+(n) = f_-(n) + \frac{n^n e^{-n}}{n!}$. Last term here approaches $0$ when $n$ approaches infinity, what means that the limit of these two will be the same.

  • @Blue: Please make sure to submit your answer before the set deadline.

  • Blue Blue

    Working on it!


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Blue Blue
  • @Blue, I think you you meant $\f_- = \frac{1}{2}$ in the last sentence.

  • @Blue, where did $f_- + f_+ = 1$ come from?

    • Blue Blue

      $\xi_n$ is either greater than or equal to $n$ or less than $n$.

  • @Blue, thats right, but $f_+ = P(\xi_n >= n), f_- = P(\xi_n > n)$ . Here the signs are "greater" and "greater or equal".

The answer is accepted.
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