How can i find, does $\lim_{n\to\infty}\frac{\eta_n}{n}$ where $\eta_n$ has poisson distribution with  $\lambda = n$ exists?
121714
$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $  meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ?    I think we can write $X(n) \sim X(1)+X(1)+\cdots+X(1)$ where the sum is taken on independent identical distribution then use the law of large number. But I am not sure that is it correct or not. Can anyone give me some hints? Thank you in advance!
24713
Let $T_1, T_2, \dots$ be iid sequence of exponential random variables with parameter $\lambda$. The sum $S_n = T_1 + T_2 + \dots + T_n$ is a Gamma distribution. Now as I understand the Poisson distribution is defined by $N_t$ as follows:  $$N_t = \max\{k: S_k \le t\}$$  How do I formally show that $N_t$ is a Poisson random variable?   Any suggestions appreciated. I tried to work out a number of proofs but cannot get to the final equation.  References
62784
As is well known, the $\mathsf{Binomial}(n,p)$ distribution converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$.  I'm pretty sure that the moments of $\mathsf{Binomial}(n,p)$ also converge to those of $\mathsf{Poisson}(a)$, but I don't know how to prove it. Convergence in distribution doesn't imply convergence of moments, in general. How can I prove that the moments converge?  I've found that the binomial probability (mass) function converges uniformly to the Poisson one. This is stronger than convergence in distribution, so perhaps it can be exploited (but if so I don't know how).
4837
Let $$a_n=\left(1-\dfrac{1}{\sqrt2}\right)\dots \left(1-\dfrac{1}{\sqrt{n+1}}\right),n\ge1$$ Then find $\lim_{n\to \infty} a_n$.   How can I proceed? I am stuck at the first step. Please help.
49414
I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
24947
When $\lim_{n\to\infty} x_n = a$, and $\lim_{n\to\infty} y_n = b$, find the limit, $$\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}.$$ Thank you for your help in advance.
70903
Given the identically distributed and independant random variables $X_1,X_2,\ldots\sim\operatorname{Po}(\lambda)$ and $S_n=X_1+\ldots+X_n$ show with induction that   $$\Pr[S_n=k]=\frac{(n\lambda)^k}{k!}e^{-n\lambda}.$$   So far for $n=1$ via the definition of the density function for Poisson:$$\Pr[S_1=k]=\Pr[X_1=k]=\frac{e^{-\lambda}\lambda^k}{k!}$$ With $n=2$ and independency: $$\Pr[X_1+X_2=k]=\sum\limits_{n=0}^k\Pr[X_1=n]\Pr[X_2=k-n]=\sum\limits_{n=0}^k\frac{\lambda^n}{n!}e^{-\lambda}\frac{\lambda^{k-n}}{(k-n)!}e^{-\lambda}$$$$=\frac{1}{k!}e^{-2\lambda}\sum\limits_{n=0}^k\binom{k}{n}\lambda^n\lambda^{k-n}=\frac{(2\lambda)^k}{k!}e^{-2\lambda}$$ I assume now that with $S_n\sim\operatorname{Po}(\lambda)$ every $X_i\sim\operatorname{Po}(\lambda/n)$ and every Poisson variable can obviously split up, but how can I prove this?
15280
Can anyone help me with this limit:  $$ \lim_{n \to \infty}{\left(e^{\frac{1}{n}} - \frac{2}{n}\right)^n}$$  I think that I should expand $e^{\frac{1}{n}}$ as the sequence $1 + \frac{1}{n} + \frac{1}{n^{2}2!} + \frac{1}{n^{3}3!} + \dots $. But I can't see how it helps me.  Edit: or maybe use logarithm?
7032
I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem?  $$ \lim_{n \to +\infty} n^{\frac{1}{n}} $$
83831
I dont get it, if I look at $$\lim \limits_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e$$    It seems like it should be $1$ because $1 \over n$ is almost zero and $1^\infty$ is still $1$
67237
Let $\{X_n\}_{n=1}^\infty$ be a sequence of random variables converging to a random variable $X$. Does this fact implies that $\lim\limits_{n\to \infty} EX_n = EX$?  If not, is there any other sufficient condition?
27664
Is there a nice limiting distribution of $\max( X_1,X_2,...,X_n) $ as $n$ goes to $\infty$, assuming that they are iid normal distributions with variance $\sigma^2$.   This is almost certainly a well known problem with a clever proof and nice solution, but I've been digging around and haven't found anything.
13687
I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me?
109896
Let's define $$P(X_n)=\begin{cases}1,&amp;w.p=\; \frac1n \\0,&amp;w.p = 1-\frac1n\end{cases}$$ I want to show that does not exist X such that :  ${\displaystyle {\overset {}{X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X.}}}$  We know that  $$ X_{n} \xrightarrow{\mathrm{a.s.}} X \iff \mathbb{P}\left(\omega:\lim_{n\to \infty}X_n(\omega)=X(\omega)\right)=1.  $$  Can you please help me with that ?
79750
Find expected value of $\frac{1}{1+\xi}$ if $\xi$ has Poisson distribution.  My try:  $$E\left[ \frac{1}{1+\xi}\right]=\sum_{k=0}^{\infty}{x}_{k}{p}_{k}=\sum_{k=0}^{\infty}\frac{1}{k+1}\frac{\lambda^ke^{-\lambda}}{k!}$$  Is it right?
43844
Hope this is a meaningful question, but I'm curious if is possible to show that   $$\lim_{n\to\infty} a^{\frac{1}{n}}=1, \text{where  }a&gt;0$$  using $\delta-\epsilon$ directly or other methods. One method that I am aware of is to use the following:     If $\{s_n\}$ is a nonzero sequence, then $\liminf\bigl|\frac{s_{n+1}}{s_n}\bigr|\le \liminf |s_n|^{\frac{1}{n}}\le \limsup |s_n|^{\frac{1}{n}}\le\limsup\bigl|\frac{s_{n+1}}{s_n}\bigr|$
16328
I need help solving this problem. I tried L'hospital and rearranging but nothing worked. $$ \lim_\limits{x→∞} x^2\left(\ln\left(1 + \frac{1}{x}\right)- \frac{1}{x+1}\right) $$