## Problem 1: ##
a = 0.5
b = 1
the.prob  = pexp(b)-pexp(a)
the.same.prob = 1-exp(-b) - (1-exp(-a))


## Problem 2a: ##
reps = 1e5
y.vec = rexp(reps)


## Problem 2b: ##

## The Slow Way:
count = 0
for ( i in 1:reps )
{
  if( (a <= y.vec[i] ) & (y.vec[i] < b) )
  {
    count = count + 1
  } 
}
count

## The Fast Way:
count = sum( (a <= y.vec ) & (y.vec < b) )
count



## Problem 2c: ##

sim.prob = count / reps
sim.prob

## The Entire Simulation:
y.vec = rexp(1e5)
sim.prob = mean( (0.5 <= y.vec ) & (y.vec < 1) )



## Problem 3: ##

## Start Vectors to store exact/simulated probs
exact.vec = NULL
sim.vec = NULL

## Generate 1e5 realizations from a standard Exponential distribution
y.vec = rexp(1e5)

## Create a vector of the left-cut points for the probs
a.vec = seq(0, 1.5, by=0.5)

for ( k in 1:length(a.vec) )
{
  a = a.vec[k]
  b = a + 0.5
  exact.vec[k] = pexp(b)-pexp(a)
  sim.vec[k] = mean( (a <= y.vec ) & (y.vec < b) )   

  ## Print the result
  cat("P(", a, "<= Y <", b,")\t", exact.vec[k], 
      "\t", sim.vec[k],  "\n", sep="")   
} 


## Problem 4: ##

## Start Vectors to store exact/simulated probs
exact.vec = NULL
sim.vec = NULL

## Generate 1e5 realizations from a standard Uniform distribution
u.vec = runif(1e5)

## Make y.vec from the u.vec
y.vec = log(1/u.vec)

## Create a vector of the left-cut points for the probs
a.vec = seq(0, 1.5, by=0.5)

for ( k in 1:length(a.vec) )
{
  a = a.vec[k]
  b = a + 0.5
  exact.vec[k] = pexp(b)-pexp(a)
  sim.vec[k] = mean( (a <= y.vec ) & (y.vec < b) )   

  ## Print the result
  cat("P(", a, "<= Y <", b,")\t", exact.vec[k], 
      "\t", sim.vec[k],  "\n", sep="")   
} 


## Coin Flipping ##

## The number of simulated games.
reps = 1e4
winnings.vector = NULL

## Games loop.
for (r in 1:reps) 
{
  ## To start a sequence of bets, set his winnings
  ## to zero
  winnings = 0
  last.flip = 0
  
  ## Start flipping until we win 2 times in a row
  while (1) 
  {
    ## Flip the fair coin 
    this.flip = 1*(runif(1) > 0.5) ## 1=Heads, 0=Tails
    
    ## Add 1 dollar to our winnings if we got heads, 
    ##  and subtract 1 if we got tails 
    winnings = winnings + 1 - 2*(this.flip == 0)
    
    ## Stop flipping when we have 2 consecutive heads
    if ((this.flip==1) & (last.flip == 1)) { break }
    last.flip = this.flip
  }  
  winnings.vector[r] = winnings
}
mean(winnings.vector)