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https://medium.com/@ro.mo.flo47/the-beta-binomial-model-an-introduction-to-bayesian-statistics-154395875f93 \ No newline at end of file diff --git a/content/math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md b/content/math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md new file mode 100644 index 000000000..8d819552f --- /dev/null +++ b/content/math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md @@ -0,0 +1,71 @@ +--- +title: Exponential Distribution +tags: + - basic + - math + - statistics + - distribution +date: 2024-06-03 +--- +# Background + +## Poisson Process + +日常生活中,大量的事件是有固定频率的。 + +- 某医院平均每小时出生3个婴儿 +- 某公司平均每10分钟接到1个电话 +- 某超市平均每天销售4包xx牌奶粉 +- 某网站平均每分钟有2次访问 + +$N(t)$等于t时间内发生事件的次数,在这个过程中,两个不重叠区间内所发生的事件数目是互相独立的随机变量,那么这个随机过程$N(t)$即是一维泊松过程。 + +# Poisson Distribution + +这个过程中,在区$[t, t+\tau]$内发生的事件数目的概率分布满足Poisson Distribution, +$$ +P[(N(t+\tau) - N(t)) = k] = \frac{e^{-\lambda\tau}(\lambda\tau)^k}{k!}, \quad k=0,1,\cdots +$$ + +$\lambda$是一个正数,为固定的参数,通常称为抵达率(arrival rate)或强度(intensity)。常常用事件在单位长度$\tau$内发生的平均频率表达。 + +Poisson Distribution可以将t代为0,简化为在接下来单位时间$\tau$内有k次时间发生的概率的分布: + +$P(N(t) = k) = \frac{e^{-\lambda t}(\lambda t)^k}{k!}$ + +# Exponential Distribution + +这些事件之间的时间间隔,是属于指数分布。 + +指数分布的公式可以从泊松分布推断出来。如果下一个事件发生要有间隔时间 t ,就等同于 t 之内没有任何事件发生。 + +$$ +P(X>t) = P(N(t)=0) = \frac{e^{-\lambda t}(\lambda t)^0}{0!} = e^{-\lambda t} +$$ + +so, + +$$ +P(X\leq t) = 1 - P(X>t) = 1 - e^{-\lambda t} +$$ + +那么指数分布的CDF即为$P(X\leq t)$,同时有: + +$$ +\text{CDF}(t) = \int_{-\infty}^{\infty} \text{PDF}(t) dt +$$ +则,可以推导出: + +$$ +\text{PDF}(t) = (1-e^{-\lambda t})' = \lambda e^{-\lambda t} +$$ + + +# Deduction + +![](math/Statistics/basic_concepot/distribution/attachments/2bbb645362366906ace3296d35612625_720.jpg) +# Reference + +* https://www.ruanyifeng.com/blog/2015/06/poisson-distribution.html +* https://zh.wikipedia.org/wiki/%E6%B3%8A%E6%9D%BE%E8%BF%87%E7%A8%8B +* https://www.le.ac.uk/users/dsgp1/COURSES/LEISTATS/poisson.pdf \ No newline at end of file diff --git a/content/math/Statistics/basic_concepot/distribution/gamma_distribution.md b/content/math/Statistics/basic_concepot/distribution/gamma_distribution.md new file mode 100644 index 000000000..086860235 --- /dev/null +++ b/content/math/Statistics/basic_concepot/distribution/gamma_distribution.md @@ -0,0 +1,62 @@ +--- +title: Gamma Distribution +tags: + - basic + - math + - statistics +date: 2024-06-03 +--- + +# Background + +## Gamma Function + +Factorial: + +$$ +n! = \prod_{i=1}^{n} i = 1\times2\times3\times\cdots\times n +$$ +那如何计算$\frac{3}{2}!$呢?通过对阶乘函数的插值将阶乘函数托展到non-integer value。但是插值的方法有很多,要如何选择合适的插值的方法? + +最重要的条件是,插值后的函数要满足阶乘最重要的条件, + +$$ +n! = n\times(n-1)! +$$ +这个插值后的广义阶乘,就是**Gamma Function**: + +$$ +\Gamma(z) = \int_{0}^{\infty} x^{z-1}e^{-x}dx +$$ +可以验算,阶乘最重要的性质并没有变,不过形式有所偏移,性质如下: + +$$ +\Gamma(z+1)=z \times \Gamma(z) +$$ +证明如下: + +![](math/Statistics/basic_concepot/distribution/attachments/prove.jpg) + +同时,在integer节点,Gamma function也和阶乘对应起来,即: + +$$ +\Gamma(n+1) = n! +$$ + +证明如下: + +![](math/Statistics/basic_concepot/distribution/attachments/df15541df80b6065fb8296d80ffceac5_720.jpg) + + + +## Exponential Distribution + +Exponential Distribution指的是,probability of the waiting time between events in a Poisson Process + +Here's the exponential distribution explain: [Exponential Distribution](math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md) + + +# Reference + +* https://www.youtube.com/watch?v=c7_F4P71E2E +* https://www.youtube.com/watch?v=GJoZWPocAm0 \ No newline at end of file diff --git a/content/math/Statistics/Basic/Quantile.md b/content/math/Statistics/basic_concepot/quantile_concept.md similarity index 100% rename from content/math/Statistics/Basic/Quantile.md rename to content/math/Statistics/basic_concepot/quantile_concept.md diff --git a/content/math/calculus/calculus_moc.md b/content/math/calculus/calculus_moc.md index c017da425..cf42336c6 100644 --- a/content/math/calculus/calculus_moc.md +++ b/content/math/calculus/calculus_moc.md @@ -15,4 +15,8 @@ date: 2024-05-24 # Integration * [Common Integration](math/calculus/integration/common_integration.md) -* [Integration by Parts](math/calculus/integration/integration_by_parts.md) \ No newline at end of file +* [Integration by Parts](math/calculus/integration/integration_by_parts.md) + +# Series + +* \ No newline at end of file diff --git a/content/math/calculus/differential/common_differential.md b/content/math/calculus/differential/common_differential.md index 4ef6b0c07..3cd81fbf3 100644 --- a/content/math/calculus/differential/common_differential.md +++ b/content/math/calculus/differential/common_differential.md @@ -7,5 +7,6 @@ tags: - math date: 2024-05-24 --- -## cos(x) +# Reference +* https://www.youtube.com/watch?v=oBlHiX6vrQY \ No newline at end of file diff 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