# 機率CH5 - Special Discrete Distributions ## 5.1 Bernoulli and Binomial Random Variables ## 白努力實驗 - 定義probability mass function - [ ] read ![](https://i.imgur.com/0FD45Y4.png) ## 白努力的期望值、變異數 - [ ] read ![](https://i.imgur.com/sSeS8vi.png) ## Binomial Random Variables 獨立事件、X參數只有n(數量).p(機率數值) - [ ] read ![](https://i.imgur.com/PqbKWQ9.png) ## 二項分布 - [ ] read ![](https://i.imgur.com/zJ0Uzfy.png) 從n個選x個成功的，順序並不重要 下圖公式裡面還需要多乘一個nCx，投影片錯誤 - [ ] read ![](https://i.imgur.com/Q6G8lJV.png) ## Ex5.8 - [ ] read ![](https://i.imgur.com/OkiMwIR.png) - [ ] read ![](https://i.imgur.com/UotfdQ2.png) ## Maximum of p(x) - [ ] read ceiling function, floor function ![](https://i.imgur.com/qxytDGA.png) ![](https://i.imgur.com/l8nJFfG.png) ![](https://i.imgur.com/6K6f89K.png) ## Binomial distribution的標準差、變異數 - [ ] read ![](https://i.imgur.com/yDTrFuL.png) ![](https://i.imgur.com/OVNIi0V.png) ## 5.2 Poisson Random Variables ## poisson random variables definition - [ ] read ![](https://i.imgur.com/IfglBSS.png) ![](https://i.imgur.com/2aRzFWC.png) ![](https://i.imgur.com/xOllEac.png) ![](https://i.imgur.com/Qmilimf.png) ![](https://i.imgur.com/sv7kYDs.png) ![](https://i.imgur.com/5siILp0.png) ## poisson process - [ ] read ![](https://i.imgur.com/1HZKB7k.png) Stationary（靜止）找到長度相同的區間，這樣的兩段分佈裡面的數量會相同。區間可以重疊 ![](https://i.imgur.com/zr3zbWS.png) Independent increment 區間不能重疊 ![](https://i.imgur.com/3m3TYnD.png) Orderliness o(h) -> f(h)趨近0的速度比h的還來得快 ![](https://i.imgur.com/pb3yXM2.png) ## Th5.2 - [ ] read ![](https://i.imgur.com/XQ8ySBw.png) proof of th5.2 ![](https://i.imgur.com/jJPLtxn.png) ![](https://i.imgur.com/CUpsBMD.png) ![](https://i.imgur.com/WyqNqad.png) binomial會在極限之下形成poisson ## poisson process with rate - [ ] read ![](https://i.imgur.com/5NUOGv0.png) ![](https://i.imgur.com/Q59Yqqt.png) ## 5.3 Other discrete random variables ## Geometric random variables - [ ] read ![](https://i.imgur.com/q42thPE.png) ## Definition of Geometric random variables - [ ] read ![](https://i.imgur.com/t1CePok.png) ## Expectation and Variances of Geometric random variables - [ ] read ![](https://i.imgur.com/3zWvyu2.png) ![](https://i.imgur.com/vmXNW1N.png) ## Memoryless property - [ ] read ![](https://i.imgur.com/ctoHDwQ.png) ## Geometric random variables is the only Memoryless discrete random variables - [ ] read ![](https://i.imgur.com/JwAYie7.png) ## Negative Binomial Distribution - [ ] read ![](https://i.imgur.com/C8JgeLO.png) ![](https://i.imgur.com/bhC4HNA.png) ## With and without replacement - [ ] read ![](https://i.imgur.com/U8nfjUL.png) ## Hyper-geometric Random Variables - [ ] read ![](https://i.imgur.com/wJ05Gid.png)