common_distributions

common_distributions - 624 Table of Common Distributions...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 624 Table of Common Distributions Discrete Distributions W Bernoulli(p) me P(X=~’Clp)=pm(1 71:01”; 3:0,1; 051331 mean and variance EX—p’ varX—pa—p) mgf MX(t)=(1—P)+P€t Wm Binomial(n, p) pmf P(X = zln,p) = (2)1230 -p)"‘\$; a: =0,1,2,...,n; o Sp 51 mean and variance EX _ np’ varX _ “pa _p) mgf Mx(t) = [pet + (1 *PH" notes Related to Binomial Theorem (Theorem 3.1.1). The multinamz'al distri- bution (Deﬁnition 4.6.1) is a multivariate version of the binomial distribution. M Discrete Uniform pmf P(X=x;N)=—;,{_,; m:1,2,...,N; N=1,2,... mean and EX = M1 VarX = gNJgiggN—Q variance 2 ’ 11 mgf MX(t) : ﬁ 2,; e“ Table of Common Distributions 625 Geometricgp) ‘ pmf P<X=a=|p)=p(1—p)I-'; w=1,2,...; ospsl 3; mean and _ l _ 1_ . ' variance EX " 19' varX _ 7’32 t r ; mgf Mxe) = t < —log(1—p) notes Y = X — 1 is negative binomialﬂ, p). The distribution is memoryless: P(X>siX>t)=P(X>s—t). Hypergeometric pmf P(X=:c[N,M,K)= “(NT ; x=0,1,2,...,K; K M—(N—K)gng; N,M,K20 mean and _ KM _, KM gN—MlgN—IQ notes If If << M and N, the range x = 0, 1,2, . . . ,K will be appropriate. Negative binomialﬂ"1 10) pmf P(X=mlr,p)=‘(T+:"’)pra—p)e w=o,1,---; ospsl ;' "’3‘?" and EX = —P—T“' ), VarX = “1— variance P mgr Mx(t)=(miw) , t<—log(1—p) E notes An alternate form of the pmf is given by P(Y = ylr,p) = (y—:)pr(1-p)y“r,y = 1",7' +1,.... The random variable Y = X +5". The 1.- negative binomial can be derived as a gamma mixture of Poissons. (See Exercise 4.34.) PoissonOU pmf P(X=e|3)=e_;,*”; z=0,1,...; 0sA<oo "‘8‘?" and EX = A, VarX = A variance mgf Mm = we“) 626 Table of Common Distributions Continuous Distributions W Beta(a, ﬂ) pdf fame) = mma‘lﬂ —z)'3"1, 05 2:3 1, a>0, ﬁ>0 3233;?" EX = W)? = mgr Mxtt) = 1+ 2:: ( :23 ﬁe) notes The constant in the beta pdf can be deﬁned in terms of gamma functions, B(a, ﬂ) = \$533. Equation (3.2.18) gives a general expression for the moments. Cauchy(6, 0*) pdf f(\$13,0)=#1—::-6-)—f,—oo<:c<oo,—oo<6<oo, cr>0 me?“ and do not exist variance mgf does not exist notes Special case of Student’s t, when degrees of freedom = 1. Also, if X and Y are independent n(0, 1), X / Y is Cauchy. Chi squared pdf ﬂme = Wrinkle—“2: 0 s w < as; p = 1,2,... : mm,” and EX 2 p, VarX = 2p variance 1 13/2 1 mgf Mxﬁ) = t t < 5 notes Special case of the gamma distribution. "1‘ Double exponentialm, a) pdf f(a:|u,cr) 2 ﬁyli—ﬂl/‘T, _00 < x < oo, —00 < ,u < 00, a > 0 mean and variance EX =u, VarX=2o2 Table of Common Distributions 627 mgf me) = itl <§ notes Also known as the Laplace distribution. Exponentialoﬂ) pdf more) = ﬁrm/’3‘, 0 s a: < oo, 5 > 0 mean and EX = ﬂ, VarX I {32 variance mgf Mx(t)=l:'§;, t<§ notes Special case of the gamma distribution. Has the memoryless property. Has many special cases: Y = XV"! is Weibull, Y : due/5 is Rayleigh, Y = a — 710g(X/}3) is Gumbel. . F I‘ ul+v2 III/2 I 2 V] x(Vl—2)/1 E v v —‘ ———,,—"';;""—‘, 0 S :12 < ; Pdf fcclylayl) 1.. (V2) (1+(é)z)( l+vgll2 00 V1, 1/2 = 1, . . . mean and _ 1/1 variance EX _ “‘2’ V2 > 2’ 2 V2 (V1+V2"2) V8IX=2(V2_2) m, V2>4 F v|+2ﬂ 92—211 moments n __ 1 2 :11 n 53. (mgfdoes not exist) EX _ F (VI) ’ n < 2 Kid 11' dF —"34 35221 11th? notes eatetoc1squaIe (1,”,2— y—I/ V2 ,were exsare _ independent) and t (Fm, = ti). Gamma(a,ﬁ) Pdf f(95|a,ﬁ) = “05,3. arm—16””, 0 S a: < 00, a. [3 > 0 mm.” and EX = a6, VarX = (If? variance .‘ a __ 1 1 mgf Mxtt) *— (Hﬂ) , t< 3 notes Some special cases are exponential (a = 1) and chi squared (a = 19/2,}3 ”—" 2)- If 04 = g, Y = x/X/ﬁ is Maxwell. Y = 1/X has the inverted gamma distribution. Can also be related to the Poissan (Example 3.2.1). 628 Table of Common Distributions ——-————————-—-—-—-—--H——n—._.__ Logisticgt, ﬂ) —z— ['6 f(\$lﬂiﬁ)mérl-_f;{7:::hr—EIEI _OO<\$<001 —00<M<°0i [3 > 0 mm.” and EX = ii, VarX = 12:91- variance mgf me) = e“‘1“(1-,6t)1“(1+ﬂt), Itl < é notes The cdf is given by F(93|,u, 6) = m. Lognormalm, 02) eeilos ac — #12 #262) 2 1 I Pdf f(\$ltt,cr)= ha I , OSm<oo, —c:.<3<1u,<ooy 0‘ > 0 3:33;? EX = ett+(02/2), VaI X = 620m?) _ 82:”ch moments I 2 _ _ eng+n 11/2 (mgf does not exist) notes Example 2.3.5 gives another distribution with the same moments. Normalm, oz) __ __ Z 2 f(xlnu',02) ; 37:03 (3: 'u) [ad )a _00 < a: < 00: —00 < ,u < 00, or > 0 me?” and EX = lu,VarX = 0’2 variance mgf Mxtt) = ema’t’ﬂ notes Sometimes called the Gaussian distribution. Pareto(o:, 6) pdf f(a:{a,ﬁ)=w%%f—., a<x<oo, a>0, ﬁ>0 meet: and EX = Egg], ﬂ >1, variance _ 02 VarX ‘* (e—liFw—Z)’ ’6 > 2 does not exist mgf Mu: . Table of Common Distributions 629 t F u—zﬂ 1 1 pdf f(.r|u)= P; m(l+(3§))w+m, moo<x<oo, 12:1,... "‘3‘?" “"d EX .—. 0, V > 1, variance VarX = V—Zz, u > 2 moments n __ F 13"] F L'2"a 11/2- (mgfdoes not exist) EX — “11%) V 1f n < u and even, EX”=0ifn<r/andodd. notes Related to F (Flay = ti). Uniform(a, 6) pdf f(mla,b)==ﬁ, :15be meanand EXﬁm VMX=M variance 2 ' 12 mgf Mm) = notes If a = 0 and b = 1, this is a special case of the beta (a = ﬂ : l). Weibullty, ﬂ) pdf ﬂash/,3) = gmv-le-IW, o s x < co, 7 > 0, r3 > o EX 2 51/71“ (1 + i), VarX = WW1“ (1 + 3-) -— 1‘2(1+§)] moments EX" 2 ﬁn“ I‘(l + g) notes The mgf exists only for '7 2 1. Its form is not very useful. A special case is exponential ('7 = 1). ...
View Full Document

This note was uploaded on 04/21/2008 for the course STAT 1211 taught by Professor Hernandez during the Spring '08 term at Columbia.

Page1 / 6

common_distributions - 624 Table of Common Distributions...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online