Anna Karlin Most Slides by Alex Tsun Poisson RV Example The Zoo of - PowerPoint PPT Presentation

Anna Karlin Most Slides by Alex Tsun

Poisson RV Example

The Zoo of Discrete RV’s ● The Negative Binomial RV ● The Bernoulli RV ● The Hypergeometric RV ● The Binomial RV probability students ● The Geometric RV Definition of ● The Uniform RV Expectation ● The Poisson RV

random variables Important Examples: Uniform(a,b): Bernoulli(p): P(X = 1) = p , P(X = 0) = 1-p μ = p, σ 2 = p(1-p) Binomial(n,p) μ = np, σ 2 = np(1-p) Poisson( λ ): μ = λ , σ 2 = λ Bin(n,p) ≈ Poi ( λ ) where λ = np fixed, n → ∞ (and so p= λ /n → 0 ) Geometric(p) P(X = k) = (1-p) k-1 p μ = 1/p, σ 2 = (1-p)/p 2 µ E EN 62 3 ! X Var X

Probability 4.1 Continuous random Variables Basics Anna Karlin Most Slides by Alex Tsun

Agenda ● Probability Density Functions (PDFs) ● Cumulative Distribution Functions (CDFs) ● From Discrete to Continuous

X IX 952 Discrete with riv range Pmt CDTE Fxwt Pr PrlX w ww.no p lw Praewnerraff Kw lw 30 p CITY.ly inIeaYTngaom EINE I to 0 FxlwtfyPxT pxlaf txlkl fxlk.it Sitt v EW is p given Ext k what

pick Fx Kt a µ EH b Px s.t.tt PrlXEk t prfxek FxCK4 FxCk D 4pxCH don't know d I rn to represent cont want suppose drew Sm Os random uniform tell OP I I In T Diseapprox r brown CDF discrete dish f B 0 WEENIE aee ee L'T w In O W O in E o.o Yei a at 11 11141 In 5 KKE I w

W eat n I Ia L pyo 1 I w h 0 p lw i n as makes sense pint longer no T density introduce ghpmffP probability font analogue pxkt fIIY txIIFXHII.hx.PT

Fx f v v CDF Intuition 1 1

CDF Intuition 1 1 o AM H 1 1

at FX f wI v CDF Intuition 0 eat 1 1 T i 1 1

prob density fn 12 1 730 I Ep a pdf XEl Properties of PDF Intuition

PDF Intuition

Delong what pdf must satisfy PDF Intuition a or

PDF Intuition e

PDF Intuition t

PDF Intuition B f g PrfX proportional g

PDF Intuition D

Probability Density Functions (PDFs) a Pra Xeb I Pr F fb Fila

Random Picture probability students Definition of Expectation

CDF Intuition 1 1

Iffy G d v CDF Intuition D 1 1 r a 1 1

w f Hdv t CDF Intuition 1 1 1 1 w

CDF Intuition 1 1 1 1

CDF Intuition 1 e 1

CDF Intuition 1 1

CDF Intuition 1 a 1

CDF Intuition 1 1

CDF Intuition 1 a 1 2

CDF Intuition 1 1

Cumulative Distribution Functions (CDFs)

OIL uniform X Model m whatisc se a c b 2y i 424 I I don't know d 4 2 0 vs f H Creve's Eo v z t f w dv I Cvf Ef Cdv O 3 E 2

F t E a b EEE v s z III a v z I don't know d

L f Cx fx x t From Discrete to Continuous f Cx Fx x EHEI fx dx EH rjP I

Probability 4.2 Zoo of Continuous RVs

Agenda ● The (Continuous) Uniform RV ● The Exponential RV ● Memorylessness

The (Continuous) Uniform RV TTT pdf T fbx.es dxEEafaba ELXKIxfxtxldx b2 atVarfX ECX7 EIT 2lb a ax ET t x x beg

The Uniform (Continuous) RV he f