Adding Day 10 Material

mc10_17.R

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+##putting together the R code for the Binomial simulation
+
+N<- 100
+M<- 50
+
+
+plot(dbinom(0:100, 100, 0.5)~c(0:100), type='h', lwd = 3, pch=20, xlab='Number', ylab='Probability')
+dev.copy(device=pdf, file='~/dropbox/teaching/pol350a/prob3/binom1.pdf', height=6, width = 6)
+dev.off()
+points(dbinom(50:100, 100, 0.5)~c(50:100), col='cornflowerblue', lwd = 3, pch=20, type='h')
+dev.copy(device=pdf, file='~/dropbox/teaching/pol350a/prob3/binom2.pdf', height=6, width = 6)
+dev.off()
+
+sum(dbinom(50:100, 100, 0.5))
+
+##let's use a simulation
+
+
+m_at_least<- c()
+sims<- 10000
+
+for(z in 1:sims){
+	out<- rbinom(1, size = 100, prob = 1/2)
+	m_at_least[z]<- ifelse(out>49, 1, 0)
+	}
+
+mean(m_at_least)
+
+
+##but now suppose that individuals' turnout decisions are 
+
+net.sim<- function(n.iters){
+	turn.net<- c()
+	turn.indep<- c()
+	for(z in 1:n.iters){
+		start<- rbinom(100, size = 1, prob = 1/2)
+		turn.indep[z]<- sum(start)
+		nets<- start[1:50]
+		for(g in 1:50){
+			nets[50 + g]<- rbinom(1, size = 1, prob = ifelse(nets[g]==1, 0.6, 0.4))
+			}
+		turn.net[z]<- sum(nets)
+		}
+	output<- list(turn.net, turn.indep)
+	return(output)
+	}
+
+first.run<- net.sim(1000)
+
+mean(first.run[[1]])
+mean(first.run[[2]])
+
+var(first.run[[1]])	
+var(first.run[[2]])	
+plot(density(first.run[[2]]), col='red', lwd = 3, main='Comparing Network, Independent')	
+lines(density(first.run[[1]]), col='blue', lwd = 3)
+text(40, 0.03, labels = 'Network', col='blue')
+dev.copy(device= pdf, file='~/dropbox/teaching/pol350a/prob3/BinomNetwork.pdf', height=6, width = 6)
+dev.off()
+
+var(first.run[[1]])	
+var(first.run[[2]])	
+
+
+
+
+
+plot(dpois(1:100, 5)~c(1:100), type='h', lwd =3 , xlab='Number Threats', ylab='Probability')
+dev.copy(device = pdf, file='~/dropbox/teaching/pol350a/prob3/PoissonExamp1.pdf', height=6, width = 5)
+dev.off()
+
+points(dpois(10:100, 5)~c(10:100), type='h', lwd =3 , col='cornflowerblue')
+dev.copy(device = pdf, file='~/dropbox/teaching/pol350a/prob3/PoissonExamp2.pdf', height=6, width = 5)
+dev.off()
+
+
+
+##analytically
+1- sum(dpois(0:9, 5))
+
+
+##via simulation
+ten_or_more<- c()
+sims<- 10000
+for(j in 1:sims){
+	ten_or_more[j]<- ifelse(rpois(1, 5)>9, 1, 0)}
+
+
+
+
+tran<- matrix(NA, nrow = 2, ncol = 2)
+tran[1,1]<- 0.8
+tran[1,2]<- 0.2
+tran[2,1]<- 0.3
+tran[2,2]<- 0.7
+
+##after one transition 
+c(1,0)%*%tran
+##after two transitions
+c(1,0)%*%(tran%*%tran)
+after three transitions
+c(1,0)%*%(tran%*%tran%*%tran)
+
+##if that continues we get 
+start<- c(1, 0)
+for(z in 1:100000){
+	start<- start%*%tran
+	}
+
+
+start_2<- c(0, 1)
+for(z in 1:100000){
+	start_2<- start_2%*%tran
+	}
+
+##the same.  	
+
+##we can also use the eigen values
+
+vec_temp<- eigen(t(tran))$vectors[,1]	
+stat<- vec_temp/sum(vec_temp)	
+
+##Page Rank!
+
+
+