common_distributions

common_distributions - 624 Table of Common Distributions...

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

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common_distributions - 624 Table of Common Distributions...

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