k<-1:W a<--1/log(1-theta) fk<-exp(log(a)+k*log(theta)-log(k)) Fk<-cumsum(fk) x<-integer(n) for(i in 1:n){ x[i]<-as.integer(sum(u [i]>Fk)) while(x[i]==N){ logf<-log(a)+(N+1)*log(theta)-log(N+1) fk<-c(fk,exp(logf)) Fk<-c(Fk,Fk [N]+fk[N+1]) N<-W+1 x[i]<-as.integer(sum(u[i]>Fk)) X+1 Code 从而产生logarithmic(0.5)分布的随机数 Previous Next First Last Back Forward 14
k<-1:N a<--1/log(1-theta) fk<-exp(log(a)+k*log(theta)-log(k)) Fk<-cumsum(fk) x<-integer(n) for(i in 1:n){ x[i]<-as.integer(sum(u[i]>Fk)) while(x[i]==N){ logf<-log(a)+(N+1)*log(theta)-log(N+1) fk<-c(fk,exp(logf)) Fk<-c(Fk,Fk[N]+fk[N+1]) N<-N+1 x[i]<-as.integer(sum(u[i]>Fk)) } } x+1 } ↓Code l ) logarithmic(0.5) ©ŸëÅÍ Previous Next First Last Back Forward 14
于Example n<-1000 theta<-0.5 x<-rlogarithmic(n,theta) #计算对数分布的分布律来对比 k<-sort(unique(x)) p<--1/log(1-theta)*theta"k/k se<-sqrt(p*(1-p)/n)#标准差 下面的结果表明样本的频率分布符合理论分布： 干Result round(rbind(table(x)/n,p,se),3) 123456789 0.7200.1850.0580.0180.0120.0030.0020.0010.001 p0.7210.1800.0600.0230.0090.0040.0020.0010.000 se0.0140.0120.0080.0050.0030.0020.0010.0010.001 Previous Next First Last Back Forward 15
↑Example n<-1000 theta<-0.5 x<-rlogarithmic(n,theta) #OéÈÍ©Ÿ©ŸÆ5È' k<-sort(unique(x)) p<--1/log(1-theta)*theta^k/k se<-sqrt(p*(1-p)/n) #IO ↓Example e°(JL² ™«©ŸŒ‹nÿ©Ÿ: ↑Result > round(rbind(table(x)/n,p,se),3) 1 2 3 4 5 6 7 8 9 0.720 0.185 0.058 0.018 0.012 0.003 0.002 0.001 0.001 p 0.721 0.180 0.060 0.023 0.009 0.004 0.002 0.001 0.000 se 0.014 0.012 0.008 0.005 0.003 0.002 0.001 0.001 0.001 ↓Result Previous Next First Last Back Forward 15
1.2.2 The Acceptance-Rejection Method 假设X和Y是随机变量，其概率函数分别为f和g.满足 f但≤e,Ytst.f因>0 9(t) 则舍选法(Acceptance-Rejection Method)可以用来生成X的随机数. 1.找一个可以方便生成随机数的随机变量Y,其概率函 数g满足f(t)/g(t)≤c,廿ts.t.f(t)>0. 2.从g中产生一个随机数y: 3.从均匀分布U(0,1)中产生一个随机数u. 4.若u<f()/(cg(),则接受x=y,否则拒绝y.重复2 4,直至产生给定个数的x. Previous Next First Last Back Forward 16
1.2.2 The Acceptance-Rejection Method bX⁄Y ¥ëÅC˛, ŸV«ºÍ©Oèf⁄g. ˜v f(t) g(t) ≤ c, ∀ t s.t. f(t) > 0 K¿{(Acceptance-Rejection Method)å±^5)§XëÅÍ. 1. Èòáå±êB)§ëÅÍëÅC˛Y , ŸV«º Íg˜vf(t)/g(t) ≤ c, ∀ t s.t. f(t) > 0. 2. lg•)òáëÅÍy. 3. l˛!©ŸU(0, 1)•)òáëÅÍu. 4. eu < f(y)/(cg(y)), K…x = y, ƒK·˝y. E2- 4, Üñ)â½áÍx. Previous Next First Last Back Forward 16
在离散型场合，对每个使得f()>0的k有 PA=PAg因=)/Lcgs&=f P(A) 1/c 对连续型随机变量场合，即需证明P(Y≤U≤织)=Fx().事实上 PW≤出，Y≤ cg(Y) 1/e PU≤器Y=w≤ -g(w)dw 1/c = f(w -g(w)dw 1-s cg(w) Fx(y) 舍选法的特点： ·优点是计算时间不随着x增加： Previous Next First Last Back Forward 17
3l—.|‹, Èzá¶f(k) > 0kk P(k|A) = P(A|k)g(k) P(A) = [f(k)/(cg(k))]g(k) 1/c = f(k) ÈÎY.ëÅC˛|‹, =Iy²P(Y ≤ y|U ≤ f(Y ) cg(Y ) ) = FX(y). Ø¢˛ P(Y ≤ y|U ≤ f(Y ) cg(Y ) ) = P(U ≤ f(Y ) cg(Y ) , Y ≤ y) 1/c = Z y −∞ P(U ≤ f(Y ) cg(Y ) |Y = ω ≤ y) 1/c g(ω)dω = c Z y −∞ f(ω) cg(ω) g(ω)dω = FX(y) ¿{A:: • `:¥OéûmÿëXxO\; Previous Next First Last Back Forward 17