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# Understand poisson distribution

IntroductionPoisson distribution can be derived from Binomial distribution when $\lim\limits_{n\to\infty}np = \lambda(\lambda\in\mathbb R)$, in which $n$ is the number of trials, $p$ is the probabilit

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# Proof of $$\lim_{x\rightarrow \infty} f(x)^{g(x)} = c^d$$

Theorem: $$c, d\in {\bf R}, \lim_{x\rightarrow \infty} f(x)=c>0, \lim_{x\rightarrow \infty} g(x) =d>0$$ then $$\lim_{x\rightarrow \infty} f(x)^{g(x)} = c^d$$ Proof: Because $y(x)=ln(x)$ is c

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# Calculate $$\lim\limits_{n\to\infty}(1-\frac{\lambda_n}n)^n$$

I came upon the calculation of $\lim\limits_{n\to\infty}(1-\frac{\lambda_n}n)^n$ when I was reviewing Poisson distribution. Notice that $\lambda_n = np_n(n\in\mathbb{Z}_{\geq 0}, 0\leq p \leq 1)$ and