chore: Update distribution titles and add reference links

content/math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md

@@ -1,5 +1,5 @@
 ---
-title: Exponential Distribution
+title: Poisson Distribution & Exponential Distribution
 tags:
   - basic
   - math
@@ -64,6 +64,7 @@ $$
 # Deduction
 
 ![](math/Statistics/basic_concepot/distribution/attachments/2bbb645362366906ace3296d35612625_720.jpg)
+
 # Reference
 
 * https://www.ruanyifeng.com/blog/2015/06/poisson-distribution.html

content/math/Statistics/basic_concepot/distribution/gamma_distribution.md

@@ -56,6 +56,57 @@ Exponential Distribution指的是，probability of the waiting time between even
 Here's the exponential distribution explain: [Exponential Distribution](math/Statistics/basic_concepot/distribution/exponential_distribution_and_poisson_distribution.md)
 
 
+# Introduction
+
+终于来到我们的主题，Gamma Distribution。
+
+在概率论和统计学中，Gamma Distribution是一种用途广泛的**双参数**连续概率分布。Exponential Distribution， Erlang Distribution和Chi Distribution是Gamma Distribution的特殊情况。
+
+Gamma Distribution可以被认为是*Exponential Distribution的extension*，相比较于Exponential Distribution only infers the probability of the waiting time for the first event, **the Gamma Distribution gives us the probability of the waiting time util the $n_{th}$ event**.
+
+## Deduction
+
+因为T时间后，时间第n次发生了，也就意味着，在时间t内，发生了n-1次事件。
+
+$$
+P(T\leq t) = 1 - P(T>t) = 1 - P(\text{0 or 1 or } \cdots \text{n-1 events in t})
+$$
+
+so,
+
+$$
+P(T\leq t) = 1 - P(T>t) = 1 - \sum_{i=0}^{n-1} \frac{(\lambda t)^{i}e^{-\lambda t}}{i!}
+$$
+means,
+
+$$
+\text{CDF}(t) = 1 - \sum_{i=0}^{n-1} \frac{(\lambda t)^{i}e^{-\lambda t}}{i!}
+$$
+so,
+
+$$
+\text{PDF}(t) = \frac{d}{dt}(1 - \sum_{i=0}^{n-1} \frac{(\lambda t)^{i}e^{-\lambda t}}{i!})
+$$
+The result:
+
+$$
+\text{PDF}(t) = \frac{\lambda e^{-\lambda t}(\lambda t)^{n-1}}{(n-1)!} = \frac{\lambda e^{-\lambda t}(\lambda t)^{n-1}}{\Gamma(n)}
+$$
+推广到一般形式
+
+$$
+\text{Gamma Distribution, } \text{PDF}(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}
+$$
+
+$\alpha$相当于之前的第个事件，再在分布中控制着分布的形状；$\beta$相当于之前的$\lambda$, 为速率参数，也是事件发生的到达率和强度；
+
+同时Gamma Distribution也有另一套等效参数$(k, \theta)$，表现为：
+
+$$
+\text{Gamma Distribution, } \text{PDF}(x) = \frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{-\frac{x}{\theta}}
+$$
+其中，$k=\alpha$, 控制着分布形状，$\beta = \frac{1}{\theta}$，控制着尺度
+
 # Reference
 
 * https://www.youtube.com/watch?v=c7_F4P71E2E