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1 Statistics & Data Analysis 2 (Part II) Zhu University EΩ={ω}ωΩ X(ω)RRX(ω) random variablex X Ω ω X(ω) x

2 (ΩF Ω P)X=X(ω) (ωω)ω xrω{ωx(ω) x} {ωx(ω) x} F Ω X(ω) X Ω X(ω) 1 2Ω ΩωX(ω) 3P(A) A={ω X(ω)<x} P(A)=P{ω X(ω)<x}x P(X x) X x

3 X(, + ) F(x)=P(X x)xcdf, cumulative distribution function X(, x]x x F(x 2 ) F(x)=P(X x) 1 F(x 1 ) 0 X - x 1 0 x 2 + F(x) 10 F(x) 1x(,+ ) 2F(x)x 1 <x 2 F(x 1 ) F(x 2 ) 3 lim Fx ( ) 0 x lim F( x) 1 x 4F(x)a lim F( x) F( a) xa0 5X(a, b]p(a<x b)=f(b)f(a) 6XaP(X=a)=F(a)F(a0)

4 2.2.2 X X Xx 1, x 2,..., x i,..., PX ( x) p i1, 2,..., x,... i i n X X(frequency function, probability mass function)x p i i 0, i 1,2,... P( X x ) p 1 i i i F( x) P( X x ) p i x x x x i i i

5 p(x) p i p 1 p 2 x 1 x 2 x i x Example 2.23X X p 1/6 1/2 1/3 1X 2P(X 0)P(-1<X 5/2)P(X>3/2)

outcome is random and can be either of two possible outcomes, success and

6 B(1, p) Bernoulli distribution X01 {X, p}={(0, 1 p), (1, p)}0<p<1 Xp(0, 1) p ( x) p p x x 0 otherwise x 1 x (1 ) 0 1 Bernoulli trial An experiment whose outcome is random and can be either of two possible outcomes, success and failure. Jacob Bernoulli ( ) Bernoulli's grave Changed and yet the same, I rise again

7 Bernoulli trial Binomial distribution nbernoullibernoullin nbernoulliap(a)=p, 0<p<1AX PX ( k) Cp(1 p) k k n k n, k=0, 1, 2,, n Xn, p X~B(n, p) n=1x~b(1, p)

8 B(n, p) B(n, p)

9 B(n, p) Example 2.24Tay-Sachs T-S1/4 441 Example p nnn p=0.1n=5

10 Geometric distribution BernoulliAP(A) = p, 0<p<1 Ak-1k X 1 P( X k) (1 p) k p k=1, 2, Xp X~G(p) k1 k1 P( X k) (1 p) p p (1 p) 1 k1 k1 k1 The probability mass function of a geometric random variable with p = 1/9

11 G(p) G(p)

12 Hypergeometric distribution P( X k), k=0, 1, 2,, n, M<N, n<n XN, M, n N 10np=M/N C C k n k M N M n C N k nk CMCNM PX ( k) Cp n (1 p) n C k k nk Example ~2 536 N Poisson X k P ( X k ) e k! k=0, 1, 2, λ>0 XλPoissonX~P(λ) np Poisson PoissonBernoulli rare eventλ

Siméon-Denis Poisson (1781 1840) La vie n est bonne qu à deux choses:

13 Siméon-Denis Poisson ( ) La vie n est bonne qu à deux choses: découvrir les mathématiques et enseigner les mathématiques. Poisson PoissonP(λ)

14 PoissonP(λ)

15 Example λ=0.61poisson Poisson

16 2.2.3 X(-, + ) f(x)xf(x) x F( x) P( X x) f( t) dt Xf(x)X probability density functiondensity function x(-, + )

17 1 2 f(x) 0, x(-, + ) f ( x) dx 1 3 a, b(-, + ), a<b, b Pa ( xb) Fb ( ) Fa ( ) f( xdx ) 4 SS PX ( S) f( xdx ) 5f(x)x P( x X xx) f ( x) x S Xxf(x)f(x) a b P( a x b) F( b) F( a) f ( xdx ) a F(x) F(b) F(a) a b

18 1F(x)(-, + ) 2f(x)x F ( x) f ( x ) 3CRP(X=C)=0 Example 2.28X 1 x e, x x F ( x ), 0 x , x 1 X

19 (uniform distribution) X 1 f ( x) b a 0,, a x b elsewhere X[a, b]x~u(a, b) f(x) 1 b a a b x 1P(X b)=p(x a)=0; 2 c, d(a, b), c<d, d 1 d c Pc ( Xd) dx c b a b a 3X 0, x a x a F ( x ), a x b b a 1, x b

20 (exponential distribution) X f ( x ) e x, x 0, 0, x 0, λ>0xλx~e(λ) E(λ)

21 E(λ) E(λ) F ( x) t PX ( t) e, ( t0) x 1 e, x 0, 0, x 0, Pt X t e e t t t1 t2 ( ), ( 0) memorylesst>0, s>0, P ( X s t X s) P( X t) 5PoissonPoisson

22 Example 2.29 Marshall 1990 nicotinic receptor suxamethoniumsuxamethonium λ1/ττ Marshall C G et al. (1990) The action of suxamethonium (succinyldicholine) as an agonist and channel blockers at the nicotinic receptor of frog muscle. Journal of Physiology, 428:

23 Gamma(Gamma distribution) X f ( x) ( ) 1 x x e x 0, x 0,, 0, α, λ>0xα, λgamma X~Γ(α, λ) Γ 1 ( ) x e x dx, 0 0 Gamma 1 ( ) x e x dx, (1) 1 1 ( ) 2 ( 1) ( ) ( n1) n! n

24 GammaΓ(α, λ) 1, 0.5 2, 0.5 3, 0.5 5, 1.0 9, 2.0 GammaΓ(α, λ) 1, 0.5 2, 0.5 3, 0.5 5, 1.0 9, 2.0

Gamma 1 1 t (, x) t e dt Gamma 2α=1, λ 3α(, shape parameter) λ (, scale parameter) 4X: 0 x F ( x) (, x / ), x 0, ( ) 0, x 0, Example 2.30UdiasRice1975 1968-1971 Gamma Gammaα=0.509, λ=0.

25 Gamma 1 1 t (, x) t e dt Gamma 2α=1, λ 3α(, shape parameter) λ (, scale parameter) 4X: 0 x F ( x) (, x / ), x 0, ( ) 0, x 0, Example 2.30UdiasRice Gamma Gammaα=0.509, λ= Gamma distribution Exponential distribution Udias A and Rice J. (1975). Statistical analysis of microearthquake activity near San Andreas Geophysical Observatory, Hollister, California. Bulletin of the Seismological Society of America, 65:

The machine consists of a vertical board with interleaved rows of pins.

26 (normal distribution) The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton to demonstrate the law of error and the normal distribution. The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.

27 X 2 ( x ) f ( x) e (- <x<+ ) 2 μ, σσ>0xμ, σ X~N(μ, σ 2 ) Bell curve A beautiful curve 1 f ( x) e 2 ( x ) 2 2 1f(x)x=μ 2x=μf(x) 3f(x)x 4x=μσf(x) 5σμf(x)Xμ σf(x) 2 1 2

28 6 σstandard deviation μmean σ P( X X ) P( 1) P( X X 2 ) P( 2) P( X X 3 ) P( 3)

29 8 ( t ) 1 x F x e dt x ( ), (, ) 2 Carl Friedrich Gauss The Prince of Mathematicians ( ) Stigler

30 μ=0, σ 2 =1N(0, 1)(standard normal distribution) 1 ( x ) e 2 2 x 2 (- <x<+ ) t x 1 2 ( x) e dt 2 2 (- <x<+ ) 1φ(-x)=φ(x) 2Φ(-x)=1-Φ(x)Φ(0)=1/2 3P(a<X b)=φ(b)-φ(a) 4P( X a) ( a) ( a) 2 ( a) 1 a>0 5P( X a) 1 P( X a) 2(1 ( a)) a>0 1 2 x 0 x

31 X~N(μ, σ 2 ) X X ~ N (0,1) 1 2 X x x PX ( x) P( ) ( ) a X b b a Pa ( Xb) P( ) ( ) ( )

32 α X~N(0, 1)φ(x)α (0<α<1) Z α α αz α PX ( Z) ( xdx ) a Z Z Z P( X Z ) ( x) dx ( x) dx ( x) dx 1 ( Z ) Z Example 2.31sonar VeithWilks 1985 Gauss Veitch J., and Wilks A. (1985). A characterization of Arctic undersea noise. J. Acoust. Soc. Amer., 77:

J Fluid Mech, 34: 497-515 33IsochoresDNA Mosaic Cohen N,

33 Example 2.32 Van AttaChen ,600PDF Van Atta C, and Chen W. (1968) Correlation measurements in grid turbulence using digital harmonic analysis. J Fluid Mech, 34: Example 2.33IsochoresDNA Mosaic Cohen N, Dagan T, Stone L and Graur D. (2005) GC composition of the human genome: in search of isochors, Mol Biol Evol, 22:

34 A Bad Example: The Bell Curve1994 IQ Intelligence exists and is accurately measurable across racial, language, and national boundaries. Intelligence is one, if not the most, important correlative factor in economic, social, and overall success in America, and is becoming more important. Intelligence is largely (40% to 80%) genetically heritable. No one has so far been able to manipulate IQ long term to any significant degree through changes in environmental factors - except for child adoption - and in light of their failure such approaches are becoming less promising. The USA has been in denial regarding these facts, and in light of these findings a better public understanding of the nature of intelligence and its social correlates is necessary to guide future policy decisions in America. Flawed assumptions, flawed methodology, bad conclusions! Economic and social correlates of IQ IQ < >125 US population distribution Married by age Out of labor force more than 1 month out of year (men) Unemployed more than 1 month out of year (men) Divorced in 5 years % of children w/ IQ in bottom decile (mothers) Had an illegitimate baby (mothers) Lives in poverty Ever incarcerated (men) Chronic welfare recipient (mothers) High school dropout

35 2.2.4 XXY=g(X) Xf(x)Y=g(X) φ(y) 1XΩ X Y=g(X)Ω Y 2yΩ Y Gy { x g( x) y} Y FY( y) P( Yy) P( g( X ) y) P( XGy) f ( x) dx Gy 3 ( y) F( y) Y Example 2.34X~N(0, 1) φ(y) Y e X

36 Xf(x)Y=g(X)y=g(x) Y=g(X)φ(y) f h( y) h( y), yy ( y) 0, elsewhere x=h(y)y=g(x)ω Y Y=g(X) X~N(μ, σ 2 )Y=aX+b Y~N(aμ+b, a 2 σ 2 ) Ω={ω}E ωωx(ω)y(ω) (X(ω), Y(ω)) n

37 (X, Y) (X, Y)joint CDF{X x, Y y} {X x}{y y} P( x1 X x2, y1 Y y2) Fx ( 2, y2) Fx ( 2, y1) Fx (, y) Fx (, y) F( x, y) P( X x, Y y) y (x, y) x 10 F(x, y) 1, F(+, + )=1 F(, y)=f(x, )=F(, )=0 2F(x, y)xy 3x 1 <x 2, y 1 <y 2 F( x, y ) F( x, y ) F( x, y ) F( x, y ) (X, Y)Xmarginal CDF F ( x) F( x, ) P( X x) X (X, Y)Y F ( y) F(, y) P( Y y) Y

38 joint PDF (X, Y)F(x, y) f(x, y)x, y x y F( x, y) f( u, v) dudv f(x, y)(x, Y) Joint probability density of the frequency of the OH stretch of HOD in liquid D 2 O and the hydrogen bond distance.

39 f(x, y) 1f(x, y) 0 2 f( x, y) dxdy F(, ) 1 3D P ( X, Y) D f( x, y) dxdy D P a X b d b, cy d f ( x, y ) dxdy c a 4x, yf(x, y) P( x X xx, y Y yy) f( x, y) xy 5F(x, y) f(x, y)(x, y) F( x, y) xy 2 f( x, y) marginal PDF conditional PDF (X, Y)f(x, y) f ( x) F ( x) f( x, y) dy X X f ( y) F( y) f( x, y) dx Y Y (X, Y)XY

40 (X, Y)f(x, y) f X (x)>0, f Y (y)>0 1 f ( x, y) f ( x y) f ( y ) ( ) x F x y f ( u y ) du Y XY=y 2 f ( x, y) f ( y x) f ( x ) ( ) y F y x f ( x v ) dv X YX=x Schematic showing joint, marginal, and conditional densities fy ( y) f( x, y) dx f ( y x) f ( x, y) f ( x) X f X ( x) f( x, y) dy

X, Yx, y PX ( xy, y) PX ( x) PY ( y) F( xy, ) F( x) F( y) XY (X, Y)XY f ( xy, ) f ( x) f( y) f(x, y), f X (x), f Y (y) X Y X Y (X, Y) 1 f( x, y) 2 2 1 x y e 2 2

41 X, Yx, y PX ( xy, y) PX ( x) PY ( y) F( xy, ) F( x) F( y) XY (X, Y)XY f ( xy, ) f ( x) f( y) f(x, y), f X (x), f Y (y) X Y X Y (X, Y) 1 f( x, y) x y e ( x ) ( xx)( yy) ( yy) x (1 ) x x y y μ x, μ y, σ x >0, σ y >0, ρ <1(X, Y)(μ x, μ y, σ x, σ y, ρ)(x, Y)~N(μ x, μ y, σ x2, σ y2, ρ) Bell curved surface

42 Example 2.35(X, Y)~N(0, 0, 1, 1, ρ) (1 ) f ( x, y) e (X, Y)f X (x), f Y (y) 2 2 x xy y Y X (X, Y)~N(μ x, μ y, σ x2, σ y2, ρ)x~n(μ x, σ x2 ) Y~N(μ y,σ y2 ) (X, Y)~N(μ x, μ y, σ x2, σ y2, ρ)xy ρ0

43 Example 2.36X~N(0, 1)Z~N(0, 1), Y=sign(X) Z sign(x) 1, x 0 sign( x) 1, x 0 Y~N(0, 1)(X, Y) Y X

Microsoft PowerPoint - Lect02 Probability_2

Microsoft PowerPoint - Lect02 Probability_2 统计与数据分析统计与数据分析 Statistics & Data Analysis 2 概率理论基础 (Part II) Zhu Huaiqiu i @Peking University 2.2 22 2 随机变量概率的概型 : 统计古典几何概型化的基本假设意义? 概率概型的局限不依赖于概型化的模型, 如何进一步研究随机事件 A 的概率? 建立一个关于随机事件 A 的数学模型! A Ω