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# System prepended metadata

title: 機率CH4 - Distribution Functions and Discrete Random Variables

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# 機率CH4 - Distribution Functions and Discrete Random Variables
# <font color="red">Use Checkbox to check whether I've read or not</font>
## EX4.6
- [x] read
區間長度固定 > 使用平移 > 寬度就會和機率大小掛鉤

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## Distribution Function
## TH4.2.1
- [x] read
![](https://i.imgur.com/yPl70ji.png)
## TH4.2.2
- [x] read
這裡會用到機率函數的連續性
Tn會包含其他Ti的聯集
![](https://i.imgur.com/zw0HSzE.png)
## TH4.2.3
- [x] read
Tn隨便跑的情況下，可能最終跑到集合之外
![](https://i.imgur.com/t06T1uQ.png)
## TH4.2.4 右連續
- [ ] read
找一個遞減的seq Ti，這個seq要有極限，且這個極限值會等於函數值![](https://i.imgur.com/L762MKO.png)
<u>seq的例子：</u>![](https://i.imgur.com/Hnt4anT.png =50%x)
## Th4.2.5 左極限
- [ ] read
從負無窮大逼近a
![](https://i.imgur.com/3PXGOPx.png)
## Summary4.2
- [ ] read
不連續點會累積機率 > 因為P(X=t)=P(t+) - P(t-). > no jumps
![](https://i.imgur.com/rG0pf1D.png)
## Remark4.1
- [ ] read
![](https://i.imgur.com/5pI1wKl.png)

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## Discrete Random Variables
## Definition4.3.1
- [ ] read
![](https://i.imgur.com/GAjmYj6.png)
## Definition4.3.2 probability mass function
- [ ] read
![](https://i.imgur.com/lw8Ihsv.png)
![](https://i.imgur.com/1pC4yKz.png =50%x)
## Ex4.12
- [ ] read
Discrete Random Variables的題目問法：
![](https://i.imgur.com/NHZBtXY.png =50%x)
![](https://i.imgur.com/uqfUiDV.png =50%x)

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## Expectations of Discrete Random Variables
## Definition4.4.1 期望值
- [ ] read
很像平均數，但在這裡叫他期望值
絕對收斂(加上絕對值之後，是否依舊收斂)那邊，這本書跟其他本書的看法不太一樣
![](https://i.imgur.com/nOyQt5s.png)
![](https://i.imgur.com/D5yEccv.png =80%x)
## Remark4.4.1
- [ ] read
![](https://i.imgur.com/0oDzz9W.png)
## Remark4.4.2 更廣義的期望值定義
- [ ] read
也就是說，這本書的定義較狹隘
![](https://i.imgur.com/sLBYVap.png)
## Ex4.4.1.9
- [ ] read
分佈相同不代表結果相同
![](https://i.imgur.com/HYBCdJR.png)
大家的分布都一樣，最大的機率就是n+1分之一
![](https://i.imgur.com/PYHcwSA.png)
## Ex - Pólya’s Urn Model 1
- [ ] read
兩種代幣放在桶子內，隨機拿一個，觀察顏色，之後放回去，不旦放回去，還多放了一些進去
Xn代表前n次抽出白色代幣的個數
這裡用歸納法來證明
1. 證明X1對的
2. 假設Xk對
3. 證明Xk+1可以回到敘述
    -    集合關係(Xn包含Xn+1)
    -    導成條件機率
    -    討論第n+1次抽取的顏色情形(抽藍(白數量k)、抽白(白數量k-1))
![](https://i.imgur.com/kiLYlhb.png)
![](https://i.imgur.com/7yrjasu.png)
![](https://i.imgur.com/SlDspMD.png)
![](https://i.imgur.com/RWxmid8.png)
![](https://i.imgur.com/gHRoZ7c.png)
![](https://i.imgur.com/zDWuKJg.png)
## Ex - Pólya’s Urn Model 2
- [ ] read
![](https://i.imgur.com/krg9vaF.png =90%x)
![](https://i.imgur.com/GcxBZDo.png =90%x)
![](https://i.imgur.com/zGv2eRK.png =90%x)
![](https://i.imgur.com/Y51EIKu.png =90%x)
## Th4.4.1
- [ ] read
![](https://i.imgur.com/kp9BRmT.png =90%x)
## Th4.4.2 Law of the Unconscious Statistician
- [ ] read
![](https://i.imgur.com/rSJWBfe.png)
pf:
(這裡g_inverse不是反函數的意思)
![](https://i.imgur.com/ae5NDka.png)
![](https://i.imgur.com/MqYcGYv.png)
![](https://i.imgur.com/qouQQGJ.png)
## Col4.4.2
- [ ] read
![](https://i.imgur.com/B5aeYTf.png)
![](https://i.imgur.com/S8C7TCt.png =90%x)
## Variances and Moments of Discrete Random Variables
## Def4.5.1 標準差、變異數
- [ ] read
![](https://i.imgur.com/QwZPm9P.png)
## Th4.5.1
- [ ] read
![](https://i.imgur.com/qpaF882.png =90%x)
## Th4.5.2
- [ ] read
![](https://i.imgur.com/z9lLyEw.png)
## Th4.5.3
- [ ] read
![](https://i.imgur.com/BGu6gM6.png)
## Ex4.5.1 Moment
- [ ] read
![](https://i.imgur.com/bwNvp6e.png =90%x)
## Remark4.5.1
- [ ] read
![](https://i.imgur.com/wl34P8p.png)
## Standardized Random Variables
## Th4.6.1
- [ ] read
![](https://i.imgur.com/FbejOo6.png)