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Distributions and Their Relationships

# Distributions and Their Relationships - Column 1 Column 2...

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Unformatted text preview: Column 1 Column 2 Column 3 Column 4 Column 5 Negative Hyper- Hypergeomelrie Iiserete Uni- geomelriewgc‘ a) (N, K, n.) form (:1. . . .‘ 1,.) a: h‘...‘N K l k 1: weather {N :{110} gative Bino- mial, Pascal (m) ex lnX 111m =111|i5a + 111(a1) r9 =1nﬁ‘n' =1fa a? =(o1aEJ‘1 9:1 —> x Row 5 Weibull a1 :1! Triangular (can) 2 ‘ (mm, b) x I_> I] a E a: 3 b a = 2 (X1+ X2);‘2. 11d a1=a2=1 azkjuteger 0:120:22]. m=[a.+b)/2 Rayleigh Exponential 3“ =1 Erlang Standard '3 = 015= 1 Rectangular w _ _ . R0 6 05') (ﬂ) “-k I (16,18) Uniform Uniform (a, b) le] m, zZﬂ_ Xhﬁd 5.20 . ogxgla+(a—a) _ _ i=1 miu{X1,...,Xn}‘iiu —,5‘ 111(1—X) X—al 1—X2, ﬁcta=a Malntéx) apt—Titltéxwm Ell—X34” i . + {RX . . ' . . Row 7 Double Exponential, Stanglartl Logistic uglogtstu: Pareto Laplace (pm?) Logistic X (m?) luX (621.622) (9,ch —x<m<x —:x:<sr<:x: i—E —x<x<x91=:g 2320 .5329 X1 — X2, 11a,a = on =1 ‘ Q2 _ 3 —1u[(X,/a)c- — 1] max {X1,X2, . . . , XN}, iicl, N N Geometric Standard Equal- .1: = 1 2 ‘1"‘ln ogalive Bino- mial, Pascal (hp 1: E Xﬁ‘ k—dim normal {:1 ,u + 0X 11111 (X — arm/1! (2,35 a—‘oo (05,15) 3 2 0 1 S (,6) Uniform z 2 O D g z 3 1 —5 Ina—X) Figure 2: Relationships among 25 Distributions Spaced Uniform (n) I iscrete Uni- f0ﬂr1[11,...‘r“] I=r1,12.-~‘In Triangular (a, m, 5) D I a = egrangular, b: 1 Uniform (a, b) _ a g z 3 b Table 1: Discrete Distributions Distribution Index Range Probability Expected Variance Name No. Mass Function Value Bernoulli R103 :1: = 0,1 pm(1 — pf”: p p[1 — p) (p) Binomial R203 :1: = 0,1,.. .,n Cgpmﬂ — p)"_‘" 7139 np(1 — p) (11119) Discrete R105 :1:: I1,5L‘2,...,1L'n 1/11 2111/11 25.63/11 Uniform 6:1 1:1 n (3:1,...,1L'n) _ (Z "Tl/n)? i=1 Equal—Spaced R205 :1: : a, a + (3, Uniform o: + 2G,. . . , b (1 + tag—1 (a + b)/2 1:28:12 — 1)/12, (a,b,c) W eren=1+b;c“ Geometric R201 :1: : 1, 2, p 1 —p ”—1 1 p 1— p p2 (was) (p) < 1 / 1 1/ Hyper— R102 :1: = CfCi‘f/Cf up up (1 — p) x: , Geometric max 11. — N + K, 0 , , _ r (N,K,n) ...,Jiiin{K,n} } “herein—Km, Negative R202 :1: = k, k +1,... Cg:llpk(1 — @934“ k/p k(1—p)/p2 Binomial (hp) . CK C”—K _ Negative R101 :1: = ,..., %:‘k kgrll W Hyper— _ N — K + k K_k__1 - (K +1 —k) Geometric ' N—zc——1 (N, K, k) Poisson R204 :1: = 0,1,2... 11“” exp(—,u)/1" ,1; 1t (11) 93—1 1 — 1 Polya R104 :1: = 0,1,2,...,n C; (p+ \$8) 7139 M (“339118) ”3:21 +18 H (1 — p + M) 1:01 i (1 +115) l—1 i=0 Note: CE, = n!/[m!(n — m)!] Table 2: Continuous Distributions Distribution Index Range Probability Expected Variance Name No. Density Function Value Arc—Sine R405 0,1] [ﬁx/\$(1 — m) ]_1 1/2 1/8 .4 W D: a a Bat-a mm 0‘1] l Iguana”) 014-1622 mm ((11, G2) C(guglfy R4C4 E { Bﬂl + [%)2] }’1 NA NA Ch' 3 d R3C5 4+ ﬁ_le_m 2 (1)1)- quare .- W U 1) 39—1 , Erlang R6C3 is." M kﬁ 9:52 (ram W43! Eggsnonential R6G2 K" ,8’1 exp(i:c/,8) ,6 £32 %(t.fe1ne R401 { 7—1 exp[—(5r: — 9V7] 9 + 0.577227 7279/6 (5;; -exp{7exp[7<x 7 9W} 141mm) R304 E rim/mum v2 — 2‘ L'1(v2—2)2(v2—4)’ ml” ﬂj—i . (1+uluglmﬁvww2 ’Ug > 2 112 > 4 I “’1 —x L a A Generalized REC-2 E” —r(::),a (ﬂannel “(1:13; 1) 13% “an; 1) Gamma . [,(£)u2] (anamﬁ) EXP 5 thfﬂli} _ F (m) T111313? R701 4 (MW epoI , Jul/:3) .u 2,52 - - €XP[-(r—#)/5] 2 2 \$51551“ R702 * W H W 5 /3 _. . El_ .2 —n2—l Loglogistrc R7G4 {4' (Ef+:xxp(—::):—“2]9 Sequin) 6[exp[7277)] ((11:92) ' csc(5) - [tan(5) — 6]csc2(5), 7L 721. where 6 — azm— 02 continuea continued Distribution Index Range Probability Expected Variance Name No. Density Function Value Lognormal R4C2 {+ (In 2n 71 mexp(oc2/2) mgexpwg) (m, a) . exp{_%[J_Llln :1“ P} - [exp(a2) — 1] EXPK’UZMEV] 2 Normal R301 E —°’ 0 (a, 02) V 21w 'u Pareto RTC5 9, oo) Gaga/Ira“ “—91, O: > 1 —l”§2—23 a > 2 (9,00 a- (0- ) (0t- ) I??? leigh R6Cl v (213/62)exp[—(a:/,8)2] WW2 52 — (ﬁg—ﬂ)? PEBClESDgUIM R605 (1, b] (b — a)‘1 (a + b)/2 (b — G)2/12 a, Standard R4C3 a [W (1+ :r.~2)]*1 NA NA Cauchy (a = 0.3 = 1) Standard R703 { L5”)? 0 «2/3 LOngt-IC l 1 JV exPl’ﬂ l (u = M? = 1} Standard R302 { Lea —a~2 2 0 1 Normal m P( / l (a = I], 0 = 1) Standard R604. 0,1] 1 1/2 1/12 Uniform ’U U T R303 E — [l 1 2 (v) F(’U/2)1/1TU 7 11> 12—2, v> 2 . (1 + airman/2 v 2 2—41! Triangular R505 [(1 b] (inﬂow—G) 7 a S 3: S m 1(11 + m + b) L{a2 + H12 W [:2 (Chmrbl “3+3:%1 m < 55' E b 3 1W8(am W ab W 7116)} Weibul] R501 R“ aﬁ_“\$“_1exp[W(r/ﬁ)°‘] 3m + a 32m W 5 (a, :6) —[,5T(1+%)]2 Note: R+ denotes [0, 00) and R denotes (—00, 00). NA denotes "not applicable". ...
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