= p x (1 p) 1 x. Var (X) =p(1 p) M X (t) =1+p(e t 1).

Size: px

Start display at page:

Download "= p x (1 p) 1 x. Var (X) =p(1 p) M X (t) =1+p(e t 1)."

Dorthy Ball
5 years ago
Views:

1 Prob. fuctio:, =1 () = 1, =0 = (1 ) 1 E(X) = Var (X) =(1 ) M X (t) =1+(e t 1) Biomial distributio Parameter: 0 1; >0; MGF: M X (t) ={1+(e t 1)}. Cosider a sequece of ideedet Ber() trials. If X = total umber of successes, the X B(, ). Probability fuctio: () = (1 ) Note: If Y 1,...,Y are uderlyig Beroulli RV s the X = Y 1 + Y Y. (same as coutig the umber of successes). Probability fuctio if: () 0, () =1. =0 =0, 1,...,, Geometric distributio This is a discrete waitig time distributio. Suose a sequece of ideedet Beroulli trials is erformed ad let X be the umber of failures recedig the first success. The X Geom(), with () =(1 ), =0, 1, 2,... 3

2 1.1.4 Negative Biomial distributio Suose a sequece of ideedet Beroulli trials is coducted. If X is the umber of failures recedig the th success, the X has a egative biomial distributio. Probability fuctio: () = (1 ), ways of allocatig failures ad successes (1 ) E(X) =, (1 ) Var (X) =, 2 M X (t) =. 1 e t (1 ) 1. If = 1, we obtai the geometric distributio. 2. Also see to arise as sum of ideedet geometric variables Poisso distributio Parameter: rate λ > 0 MGF: M X (t) =e λ(et 1) Probability fuctio: () = e λ λ, =0, 1, 2,...! 1. The Poisso distributio arises as the distributio for the umber of oit evets observed from a Poisso rocess. Eamles: Figure 1: Poisso Eamle Number of icomig calls to certai echage i a give hour. 4

3 2. The Poisso distributio also arises as the limitig form of the biomial distributio:, 0 λ The derivatio of the Poisso distributio (via the biomial) is uderied by a Poisso rocess i.e., a oit rocess o [0, ); see Figure 1. AXIOMS for a Poisso rocess of rate λ > 0 are: (A) The umber of occurreces i disjoit itervals are ideedet. (B) Probability of 1 or more occurreces i ay sub-iterval [t, t+h) is λh+o(h) 0) (aro rob. is equal to legth of iterval λ). (h (C) Probability of more tha oe occurrece i [t, t + h) is o(h) is small, egligible). (h 0) (i.e. rob Note: o(h), roouced (small order h) is stadard otatio for ay fuctio r(h) with the roerty: r(h) lim h 0 h =0 "! % # $!%!# $ # % &!#!% Figure 2: Small order h: fuctios h 4 (yes) ad h (o) 5

4 1.1.6 Hyergeometric distributio Cosider a ur cotaiig M black ad N white balls. Suose balls are samled radomly without relacemet ad let X be the umber of black balls chose. The X has a hyergeometric distributio. Parameters: M,N > 0, 0 < M + N Possible values: ma (0, N) mi (, M) Prob fuctio: M N () = M + N, E(X) = M M + N, + N MN Var (X) =M M + N 1 (M + N). 2 The mgf eists, but there is o useful eressio available. 1. The hyergeometric distributio is simly # samles with black balls, # ossible samles M N = M + N. 2. To see how the limits arise, observe we must have (i.e., o more tha samle size of black balls i the samle.) Also, M, i.e., mi (, M). Similarly, we must have 0 (i.e., caot have < 0 black balls i samle), ad N (i.e., caot have more white balls tha umber i ur). i.e. N i.e. ma (0, N). 3. If we samle with relacemet, we would get X B, = M M+N. It is iterestig to comare momets: 6

5 fiite oulatio correctio hyergeometric E(X) = Var (X) = M+N [(1 )] M+N 1 biomial E() = Var (X) = (1 ) whe samle all balls i ur Var(X) 0 4. Whe M, N >>, the differece betwee samlig with ad without relacemet should be small. Figure 3: = 1 3 If white ball out Figure 4: = 1 2 (without relacemet) Figure 5: = 1 3 (with relacemet) Ituitively, this imlies that for M,N >>, the hyergeometric ad biomial robabilities should be very similar, ad this ca be verified for fied,, : lim M,N M M + N N 0 10 A@ M + N 1 A 1 A = (1 ). 7

6 1.2 Cotiuous Distributios Uiform Distributio CDF, fora<<b: that is, F () = f() d = a 0 a 1 b a d = a b a, a F () = b a a<<b 1 b Figure 6: Uiform distributio CDF M X (t) = etb e ta t(b a) A secial case is the U(0, 1) distributio: 1 0 <<1 f() = 0 otherwise, F () = for 0 <<1, E(X) = 1 2, Var(X) = 1 12, M(t) 1 =et. t 8

7 1.2.2 Eoetial Distributio CDF: F () = 1 e λ, M X (t) = λ, λ > 0, 0. λ t This is the distributio for the waitig time util the first occurrece i a Poisso rocess with rate arameter λ > If X E(λ) the, P (X t + X t) =P (X ) (memoryless roerty) 2. It ca be obtaied as limitig form of geometric distributio Gamma distributio with f() = λalha Γ(α α 1 e λ, α > 0, λ > 0, 0 gamma fuctio : Γ(α) = 0 t α 1 e t dt α λ mgf : M X (t) =,t<λ. λ t Suose Y 1,...,Y K are ideedet E(λ) radom variables ad let X = Y 1 + +Y K. The X Gamma(K, λ), for K iteger. I geeral, X Gamma(α, λ), α > α is the shae arameter, λ is the scale arameter Note: if Y Gamma (α, 1) ad X = Y, the X Gamma (α, λ). That is, λ is λ scale arameter. 9

8 Figure 7: Gamma Distributio 2. Gamma 2, 1 distributio is also called χ 2 (chi-square with df) distributio 2 if is iteger; χ 2 2 = eoetial distributio (for 2 df). 3. Gamma (K, λ) distributio ca be iterreted as the waitig time util the K th occurrece i a Poisso rocess Beta desity fuctio Suose Y 1 Gamma (α, λ),y 2 Gamma (β, λ) ideedetly, the, Remark: see soo for derivatio! X = Y 1 Y 1 + Y 2 B(α, β), Normal distributio X N(µ, σ 2 ); M X (t) =e tµ e t2 σ 2 / Stadard Cauchy distributio Possible values: PDF: f() = 1 π R 1 ; (locatio arameter θ = 0) 1+ 2 CDF: F () = π arcta E(X), Var(X),M X (t) do ot eist. 10

9 Figure 8: Beta Distributio the Cauchy is a bell-shaed distributio symmetric about zero for which o momets are defied. Figure 9: Cauchy Distributio (Poitier tha ormal distributio ad tails go to zero much slower tha ormal distributio.) If Z 1 N(0, 1) ad Z 2 N(0, 1) ideedetly, the X = Z 1 Z 2 Cauchy distributio. 11

ENGI 4421 Discrete Probability Distributions Page Discrete Probability Distributions [Navidi sections ; Devore sections

ENGI 4421 Discrete Probability Distributions Page Discrete Probability Distributions [Navidi sections ; Devore sections ENGI 441 Discrete Probability Distributios Page 9-01 Discrete Probability Distributios [Navidi sectios 4.1-4.4; Devore sectios 3.4-3.6] Chater 5 itroduced the cocet of robability mass fuctios for discrete