Random Variables

Size: px

Start display at page:

Download "Random Variables"

投寓苗
4 years ago
Views:

1 Random Variables W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

2 Song 歌詞 2 樣本點, 實數值, 隨機變數結連理 3 cdf, pdf, 身份證件辨唯一 4 期望值, 變異數, 又名一二階動差 5 馬可夫, 柴比雪夫, 神龍現尾真稀奇 W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

3 Practice Problems. List two well-known random variables ( 有名字的隨機變數 ): and. 2. If we know, we can compute the probability of any event relating to the random variable X. 3. 解釋歌詞意思樣本點, 實數值, 隨機變數結連理 cdf, pdf, 身份證件辨唯一期望值, 變異數, 又名一二階動差馬可夫, 柴比雪夫, 神龍現尾真稀奇 W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

4 Random Variable (Song) 隨機變數基本概念 ( 三輪車調 ) 樣本點, 實數值, 隨機變數結連理 cdf, pdf, 身份證件辨唯一期望值, 變異數, 又名一二階動差馬可夫, 柴比雪夫, 神龍現尾真稀奇 sample point, real number, random variable connects them cdf. pdf, ID the role they play E(X), V(X), st and 2nd Moments Markov, Chebyshev, P of tail events so easy. 作業是熟讀本章的內容, 然後修改歌詞以能闡釋該章核心精神為評分標準 W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

5 樣本點, 實數值, 隨機變數結連理 E. Flip a fair coin Outcomes H or T Compute P( H ) X: s r,s S,r R S T H Sample space: S= { H, T } X(T) = 0 0 X(H) = R Q: List one Advantage of Using the R.V. X A: Quantitative ( 量化 ) W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

6 Epress Events via R.V. Event, E Event P(E) P(E) Type ( 完整 ) ( 縮寫 ) ( 完整 ) ( 縮寫 ) {s : X(s) } {X } P({X }) P(X ) 2 {s : X(s) < } {X < } P({X < }) P(X < ) 3 {s : X(s) } {X } P({X }) P(X ) 4 {s : X(s) = } {X = } P({X = }) P(X = ) 5 {s : X(s) > } {X > } P({X > }) P(X > ) 6 {s : a X(s) b} {a X b} P({a X b}) P(a X b) 7 {s : a X(s) < b} {a X < b} P({a X < b}) P(a X < b) 8 {s : a < X(s) b} {a < X b} P({a < X b}) P(a < X b) 9 {s : a < X(s) < b} {a < X < b} P({a < X < b}) P(a < X < b) Q: List more advantages of Using a Random Variable A: () Quantitative ( 量化 ) A: (2) Simplify the epression for events (e. Flip a fair coin twice, event that one H appears: {(T,H), (H,T)}. Can we simply this epression? A: (3) All events can be epressed as one of the above 9 types or their combinations W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

7 隨機變數 X, 證件, 屬性 Identification (ID, 證件 ) cdf: F X () = P(X ) pmf or pdf: F X () F X ( ) = P(X = ), 若 X 為離散型隨機變數 f X () = df X (), 若 X 為連續型隨機變數 d mgf: M X (t) = E(e t ) Characteristics ( 屬性 ) Moments ( 動差 ): Epected value, variance, skewness, and kurtosis,... quartiles ( 四分位數 ) Percentiles ( 百分位數 ) Tail events probability ( 尾端機率 ) : Markov and Chebyshev s Inequalities 將有關隨機變數的課題分成兩類 : 證件與屬性, 你的看法呢? W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

8 ID : F X (), cdf of X Definiton of F X () F X () = P(X ) type P(E) Via F X () P(X ) F X () 2 P(X < ) F X ( ) 3 P(X ) = P(X < ) F X ( ) 4 P(X > ) = P(X ) F X () 5 P(X = ) = P(X ) P(X < ) F X () F X ( ) 6 P(a X b) = P(X b) P(X < a) F X (b) F X (a ) 7 P(a < X b) = P(X b) P(X a) F X (b) F X (a) 8 P(a X < b) = P(X < b) P(X < a) F X (b ) F X (a ) 9 P(a < X < b) = P(X < b) P(X a) F X (b ) F X (a) Q: Why can F X () play the role of ID? A: With F X (), we can compute the probability of any event relating to X W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

9 ID 2: f X (), pmf or pdf of X Definiton of f X () F X () F X ( ) = P(X = ), 若 X 為離散型隨機變數 f X () = df X (), 若 X 為連續型隨機變數 d type P(E) discrete continuous P(X ) y f X (y) f X (y)dy 2 P(X < ) y< f X (y) f X (y)dy 3 P(X ) y< f X (y) f X (y)dy 4 P(X > ) y f X (y) f X (y)dy 5 P(X = ) f X () 0 (why?) 6 P(a X b) f X (y) ba f X (y)dy a y b 7 P(a < X b) f X (y) ba f X (y)dy a<y b 8 P(a X < b) f X (y) ba f X (y)dy a y<b 9 P(a < X < b) f X (y) ba f X (y)dy a<y<b A: With f X (), we can compute the probability of any event W. M. Song relating 桑慧敏 Tsingto HuaXUniv. 清華大學 Random Variables / 6

10 F X () and f X () discrete R.V.s f X () = F X () F X ( ) = P(X = ) F X (a) = a f X() continous R.V.s f X () = df X() d F X (a) = a f X()d F X () (a) cdf f X () (b) pmf Discussion. The use of plots F X () (a) cdf f X () (b) pdf W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

11 Match F X () with f X () f X () 0.8 F X () f X () (a) () F X () f X () (b) (2) F X () (c) (3) Match: (a, 2), (b, 3), (c, ) W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

12 Match F X () with f X () (conti.) f X () F X () 0 0 (a) () f X () F X () 0 0 (b) (2) f X () F X () 0 0 (c) (3) Match: (a, 3), (b, ), (c, 2) W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

13 期望值, 變異數, 又名一二階動差 Characteristics ( 屬性 ) of random variables Moments ( 動差 ): E(X k ) or E[(X µ X ) k ]; Epected value, variance, skewness, and kurtosis,... quartiles ( 四分位數 ): q i = P(X q i ) 0.25i, i =, 2, 3 Percentiles ( 百分位數 ) Tail events probability ( 尾端事件機率 ) : Markov and Chebyshev s Inequalities Process capacity measures ( 製程能力指標 ): C p, C pk,... (tet 8.3) W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

14 馬可夫, 柴比雪夫, 神龍現尾真稀奇 Markov s Inequality: Suppose that Y is a non-negative random variable and E(Y) <. For any positive real number a > 0, we have P(Y a) E(Y) a Chebyshev s Inequality: Let X be any random variable with E(X) <, σx 2 <. For any k > 0, P( X µ X kσ X ) /k 2 With Markov s and Chebyshev s Inequality, but without cdf or pdf, we can get the probability upper bounds of some events. W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

15 Well-Known Random Variables Reference: Relationships among 25 Distributions (Song, W.-M. T. and Chen, Yi-chun (20), Eighty Univariate Distributions and their Relationships Displayed in a Matri Format, IEEE Transactions on Automatic Control, Vol. 56, No. 8, pp ) W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

16 More questions. 桑老師試作的隨機變數 ( 三輪車調 ), 大家試著唱唱看, 並想一想此歌詞是否有幫助你更了解本章涵意? 2. 舉例說明隨機變數的功用? 3. 累積分配函數 F X () 與分配函數 f X () 之關係為何? 分別以公式與圖形說明之 4. 符號 F X () 與 f X () 中下標示大寫, 括號內是小寫, 為何? 5. 累積分配函數與分配函數既然有一對一的關係, 何時特別需要使用分配函數? 何時特別需要使用累積分配函數? 舉例說明之 6. 列出一個有名字的隨機變數 (hint: 如果隨機變數 X 服從常態分配, 則稱隨機變數 X 為常態隨機變數 ) 7. If we know cdf F X, or pdf f X, we can compute the probability of any event relating to the random variable X. 參考書 : 桑慧敏, 機率與統計原理第三章, McGrawHill 出版, 2007 W. M. Song 桑慧敏 Tsing Hua Univ. 清華大學 Random Variables / 6

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University 508 YiFu Lou talk 06/

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University hjzhang001@gmail.com 508 YiFu Lou talk 06/04/2010 - Page 1 Outline 508 YiFu Lou talk 06/04/2010