6 d. c9 L6 E3 z$ z Q 9 t# N6 A9 p- M# x/ U& E% x I
4 & b# h; P5 O2 r, N1 G 4 M9 E8 `6 a: K , h, B$ \) i, Z1、几何分布 ; ^4 T3 D. M0 Q( a: ^ G– 密度函数:f ( x ) = ( 1 − p ) ( x − 1 ) p , f(x) = (1-p)^{(x-1)}p,f(x)=(1−p) , j G: D7 U# B6 u& l
(x−1)3 m/ ?) \/ G% B1 A* p( R" M, X
p, x = 1 , 2 , 3 , . . . . . . x = 1,2,3, ... ...x=1,2,3,...... 1 Z! S! B% S8 F- W9 w3 B# a! e. W$ r9 O
# U [. R0 ?' U+ O: ?) N
– 分布函数:F ( x ) = ∑ k = 1 x f ( k ) = 1 − ( 1 − p ) x F(x) = \sum_{k=1}^x f(k) = 1 - (1-p)^xF(x)=∑ 2 |( m3 \6 w5 R- r6 ~k=1: `( l8 u/ o& B& J
x2 @7 s' {2 u8 @$ Z5 \# }4 {
7 X+ A, u/ l5 C: w& k! y f(k)=1−(1−p) 4 u' R* P; g9 a9 K# ]4 Z; e
x `& ~ G' V5 l& ~! S
& o5 v7 x5 n+ Q* W5 Z
0 G. Z$ E. y& t* [
; @7 \. j$ R4 ~; d# N% N
– 期望:E ( X ) = ∑ k = 1 x k f ( k ) = 1 p E(X) = \sum_{k=1}^x kf(k) = \frac{1}{p}E(X)=∑ " v9 A- E# E+ l# ?4 S+ |6 g% Gk=1 4 h% |: V! B+ qx8 e5 N4 u5 u+ ~9 i$ l
& _: q/ Q1 H& ~0 d* B kf(k)= 9 x8 `/ d1 D! p
p 9 e3 C0 | l3 z. C" h1 x0 u: ?; S18 T3 E1 {) P) h# r. o1 t
* z6 ?) h$ n9 z 8 h7 P7 g8 r1 T4 t * m8 y9 f2 ~3 V9 w # S/ k+ m4 Q7 Q1 q– 方差:D ( X ) = ∑ k = 1 x k 2 f ( k ) − E ( X ) 2 = 1 − p p 2 D(X) = \sum_{k=1}^x k^2f(k) -E(X)^2= \frac{1-p}{p^2}D(X)=∑ : y8 }8 d: H6 c5 J
k=1 1 |; ]+ @6 l! h. @1 dx - }' n, z. V7 P, T$ q 6 s4 T( c) O8 v8 F5 e, I4 f7 l k ! A" q7 C6 U0 w( ` g# B, m
2 + ?2 y4 u& T- U7 o! @' \" }) G f(k)−E(X) 0 O A8 ]1 R8 p; R# y( y25 I0 A6 O3 H% ] O
= ; E u6 F. B" U7 pp , `# j" P* E) Q: q& G5 J0 x, s# }2 3 r6 f* r f+ U+ B, z: |# s - }5 D5 W- L' ]- B2 ~" D
1−p 9 E. L8 f. Y: q8 s & A; [/ }* Z! O X
0 j! P0 } @( ? N( N1 x; x" J, j2 }* e! L
" i& Y2 M' }4 n0 o2 F% R– 矩母函数:M ( t ) = p e i t 1 − ( 1 − p ) e i t M(t) = \frac{pe^{it}}{1-(1-p)e^{it}}M(t)= / C4 C& E& ~. |) {' c% r1−(1−p)e . v" f2 G2 }' Rit ! h& R8 L: i# ?, D- g6 X. s0 b + r$ ]0 ]" t7 r `- L
pe 3 l' I! U' N4 ?& x$ i* C+ k0 k
it + v* N0 K H* R% @: e1 S1 O * J# v6 e d- E# f3 N
3 n# S* y0 ^+ Z& K 3 Z& @6 r1 ?( j/ d/ w/ p n# _. c $ x) z! _* p5 i; } Y; a4 ~ ! ^0 R! n, \+ ~# l9 S4 ]! j& q– 偏度:S k e w ( X ) = 2 ( 1 − p ) 1 / 2 Skew(X) = 2(1-p)^{1/2}Skew(X)=2(1−p) # Y. R( {* ]* `4 O. n1/20 | O. e7 p9 T+ L! {4 ^
3 u5 x' M! R" ~8 c0 ]* G* v7 Q
/ P8 g7 J) S4 G8 P( O9 k
– 峰度:k u r t ( X ) = 9 − 6 p kurt(X) = 9-6pkurt(X)=9−6p , ~& J1 ?8 f* d6 P ; c! c" }+ A" _0 U& |8 e% A" y. ~/ _: |# A
函数 功能! Q6 |7 I' z5 e( m# L+ Z
dgeom(x, prob, log = FALSE) 概率密度 5 I# _/ o% V$ {. Ypgeom(q, prob, lower.tail = TRUE, log.p = FALSE) 累计密度! X9 F6 Q0 w' K+ S1 S: W
qgeom(p, prob, lower.tail = TRUE, log.p = FALSE) 分位数 : Z+ U. r8 f, e5 l' |rgeom(n, prob) 随机数 & }+ O* P4 Z. \ X8 j. }- Z几何分布的各中心距来自5: Q0 o( X1 R* n! D1 g/ t ! d$ l7 [" K2 w! {6 U7 W6 ?- _; {* l) @' c5 X+ T
5 j5 R* ]" i% k; G+ ]1 o 2 [. d1 R+ o# g" E2、负二项分布/ s% e9 A9 ~; J+ g. Y
– 矩母函数:M ( t ) = ( 1 − p ) r ( 1 − p e t ) − r M(t) = (1-p)^r(1-pe^t)^{-r}M(t)=(1−p) $ S( {6 b" ^) [ I$ Q7 `r ( s0 F' i! ^( h. i) r (1−pe ! _/ s I; W I2 V- o$ ht/ |( S) }% U: S1 p# y6 J2 U
) " X: h; Y, u: ^9 K−r' c1 m5 N. {: T" [8 x
o: S# w5 W+ Z3 B0 B! j+ w; t$ @% |& T/ H0 X
0 k. F/ D; A6 D2 Q4 Z; b3 }– 偏度:S k e w ( X ) = n 3 + 3 n 2 + 2 n − ( 3 n 2 + 3 n ) p + n p 2 ( n 2 + n ( 1 − p ) ) 3 / 2 Skew(X) = \frac{n^3+3n^2+2n-(3n^2+3n)p+np^2}{(n^2+n(1-p))^{3/2}}Skew(X)= * C1 ~. w/ d! Z2 I5 T! A% C(n ( o9 Z" M# v ~/ e. p9 z0 X! Q; x: z
2- m" V$ l P! [' a
+n(1−p)) 8 [; J/ O/ _$ I# W i; ?
3/2$ S( _ c& B, O6 Q# o/ ?$ N
1 J9 i; Z3 o$ Mn : u" R$ M* l- `
3. l( m$ s& P+ [) i8 D$ }
+3n # N/ S& |+ H! M" F9 b6 `1 r
2. _' }- d3 \+ Z" ~% p; w
+2n−(3n 8 Z- P+ T0 |# S3 i! Q+ y22 c& D/ I9 A K% a( r4 N8 C }4 w
+3n)p+np + b$ C& z; L+ J2 4 h8 v( c& c, j3 {5 V$ L G2 t " d8 p# m9 ~' D. Y - h/ y" K% b- J, W& H; I % W. t- r! J& a" H z/ `, S) V : A7 p9 h. Z! f; b" ^: q) {6 C- ~0 D/ U* W
– 峰度:k u r t ( X ) = 略 kurt(X) = 略kurt(X)=略 (带入递推公式自行运算) x% [* v/ V5 S5 Z- j