lab 06

lab06.R

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+# 1a.
+SimDiceSample = function(m,n)
+{
+  threes<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    rolls<-sample(c(1,2,3,4,5,6), m, replace=T)
+    threes[i]<-sum(rolls==3)
+  }
+  
+  freq<-table(threes)/n
+  print(freq)
+  
+  barplot(freq,
+          main=paste("Probability histogram for", m, "dice - Sample()"),
+          xlab="Number of 3's",
+          ylab="Probabilities")
+  
+  return(freq)
+}
+
+# 1b.
+SimDiceBinom = function(m,n)
+{
+  threes<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    threes[i]<-rbinom(1,m,1/6)
+  }
+  
+  freq<-table(threes)/n
+  
+  print(freq)
+  
+  barplot(freq,
+          main=paste("Probability histogram for", m, "dice - rbinom()"),
+          xlab="Number of 3's",
+          ylab="Probabilities")
+  
+  return(freq)
+}
+
+SimDiceSample(10, 10000)
+SimDiceBinom(10, 10000)
+
+
+# 3.
+dbinom(100,300,1/3)
+pbinom(100,300,1/3)
+pbinom(99,300,1/3)
+1-pbinom(109,300,1/3)
+pbinom(110,300,1/3)-pbinom(89,300,1/3)
+
+# 4.
+x<-0:4
+p<-dhyper(x,4,48,8)
+cbind(x,p)
+
+barplot(p,
+        names.arg=x,
+        ylab="Probabilities",
+        main="Probability histogram: # of jacks in 8 cards")
+
+# 5a.
+SimJacksSample = function(m,n)
+{
+  deck<-c(rep(1,4),rep(0,48))
+  jacks<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    draws<-sample(deck,m,replace=F)
+    jacks[i]<-sum(draws==1)
+  }
+  
+  freq<-table(jacks)/n
+  print(freq)
+  
+  barplot(freq,
+          ylab="Relative frequency",
+          main=paste("Simulation (sample): # of jacks in",m,"cards"))
+  
+  return(freq)
+}
+
+SimJacksSample(8,10000)
+
+# 5b.
+SimJacksHyper = function(m,n)
+{
+  jacks<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    jacks[i]<-rhyper(1,4,48,m)
+  }
+  
+  freq<-table(jacks)/n
+  print(freq)
+  
+  barplot(freq,
+          ylab="Relative frequency",
+          main=paste("Simulation (rhyper): # of jacks in",m,"cards"))
+  
+  return(freq)
+}
+
+SimJacksHyper(8,10000)
+
+# 6.
+dhyper(100,333,666,300)
+phyper(100,333,666,300)
+phyper(99,333,666,300)
+1-phyper(109,333,666,300)
+phyper(110,333,666,300)-phyper(89,333,666,300)
+
+# 8.
+p<-1/5
+x<-0:100
+prob<-dgeom(x,p)
+
+keep<-prob>=0.0001
+x2<-x[keep]
+prob2<-prob[keep]
+
+cbind(x2,prob2)
+
+barplot(prob2,
+        names.arg=x2,
+        ylab="Probabilities",
+        main="Exact distribution: # losing tickets before first win (p=1/5)")
+
+
+# 9a.
+SimLotterySample = function(n,p)
+{
+  losses<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    count<-0
+    repeat
+    {
+      draw<-sample(c("W","L"),1,replace=T,prob=c(p,1-p))
+      if (draw=="W") break
+      count<-count+1
+    }
+    losses[i]<-count
+  }
+  
+  freq<-table(losses)/n
+  print(freq)
+  
+  barplot(freq,
+          ylab="Relative frequency",
+          main="Simulation (sample): # losing tickets before first win")
+  
+  return(freq)
+}
+
+SimLotterySample(10000,1/5)
+
+# 9b.
+SimLotteryGeom = function(n,p)
+{
+  losses<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    losses[i]<-rgeom(1,p)
+  }
+  
+  freq<-table(losses)/n
+  print(freq)
+  
+  barplot(freq,
+          ylab="Relative frequency",
+          main="Simulation (rgeom): # losing tickets before first win")
+  
+  return(freq)
+}
+
+SimLotteryGeom(10000,1/5)
+
+# 11
+p<-0.001
+
+pgeom(499,p)
+1-pgeom(1198,p)
+pgeom(1999,p)-pgeom(998,p)
+
+# 12.
+lambda<-0.75
+x<-0:30
+p<-dpois(x,lambda)
+
+keep<-p>=0.0001
+x2<-x[keep]
+p2<-p[keep]
+
+cbind(x2,p2)
+
+barplot(p2,
+        names.arg=x2,
+        ylab="Probabilities",
+        main="Exact distribution: # flaws per meter (lambda=0.75)")
+
+# 13.
+SimFlaws = function(n,lambda)
+{
+  flaws<-c(1:n)
+  
+  for (i in 1:n)
+  {
+    flaws[i]<-rpois(1,lambda)
+  }
+  
+  freq<-table(flaws)/n
+  print(freq)
+  
+  barplot(freq,
+          ylab="Relative frequency",
+          main=paste("Simulation: # flaws per meter (lambda =",lambda,")"))
+  
+  return(freq)
+}
+
+SimFlaws(10000,0.75)
+
+# 15.
+lambda<-34
+
+dpois(34,lambda)
+ppois(30,lambda)
+ppois(29,lambda)
+1-ppois(38,lambda)
+1-ppois(37,lambda)
+ppois(40,lambda)-ppois(29,lambda)
+
+(ppois(34,lambda))^2
+ppois(68,2*lambda)
+