2016-10-06

The empty set is the subset of any set

The empty set is the subset of any set. This is true. Some people take it as a convention, but in fact, it can be explained. According to the definition of $\subset$, $\varnothing\subset A\Leftrighta

Math

2016-10-01

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

Math

2016-10-01

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 $y(x)=ln(x)$ i

Math

2016-10-01

Proof of the limit's composition law when x approaches infinity

Theorem: If $f(x)$ is continous at $b$ and $\lim\limits_{x \to\infty}g(x) = b$, then $$\lim\limits_{x \to \infty}f(g(x)) = f(b) = f(\lim\limits_{x\to \infty}g(x))\tag1$$ Proof: Because $f(x)$ is con

Math

2016-09-30

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 $

Category: Math

The empty set is the subset of any set

Understand poisson distribution

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

Proof of the limit's composition law when x approaches infinity

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