outcomes <- 1:6
outcomes

toss <- sample(outcomes,size=2,replace=T) # draw a sample of 2 with replacement from outcomes
toss
sum(toss)  # compute the sum 

n <- 10000 # number of simulated tosses of 2 dice

sums <- rep(0,n) # create a vector to store the simulation results
sums
# loop to repeat the experiment of sampling with replacement from outcomes

for (i in 1:n) {
	toss <- sample(outcomes,size=2,replace=T)
	sums[i] <- sum(toss)
}
sums 

hist(sums)   # simple histogram showing frequency
hist(sums,freq=F)  # simple histogram showing relative frequency

s <- seq(from=1.5,by=1,to=12.5)
hist(sums,breaks=s,freq=F) # better histogram 

hist.save <- hist(sums,breaks=s)  # hist.save is an object with 8 attributes
print(hist.save)
empirical <- hist.save$density # relative frequencies of occurence for each of the 11 possible sums

theoretical <- c(1,2,3,4,5,6,5,4,3,2,1)/36  # theoretical probabilities assuming an equally likely sample space
cbind(empirical,theoretical)  # compare probabilities

plot(empirical~theoretical,pch=16,xlab="Theoretical probability",ylab= "Empirical probability")
abline(a=0,b=1) # add a line with intercept 0 and slope 1
############## Homework problem for Friday September 18 ###########

#  Consider tossing a four-sided object with sides labeled 1,2,3,4. Assume that 
#  each side is equally likely to be face-down after a toss, and that the down-side number is observed. 
#  Suppose that the object is tossed 3 times.
#  1. Determine the theoretical distribution of the sum of the 3 tosses. 
#  2. Modify the script above and compute an empirical distribuion. Compare the distributions. 
#  Verify that they are similar. If not, one or both distribbutions are incorrect. 
# Turn in: the theoretical probability distribution of X = "sum of the 3 down-sides", a histogram of the 
# empirical distribution, and a plot of the theoretical vs empirical probabilities.

##########################################################################################

## 	compute an approximation of the expected value of the sum of 2 dice and compare it to the theoretical value 
values <- 2:12
theoretical.expected.value <- sum(values*theoretical)
c(mean(sums),theoretical.expected.value)

# Compute variance and std deviation of the sum

variance <- sum((values^2)*theoretical) - theoretical.expected.value^2
std <- sqrt(variance)
c(variance,std)

# compare to sample variance and standard deviation
c(var(sums),sd(sums))

# WARNING: var(sums) and sd(sums) are estimators of variance
# and standard deviation, and rarely agree exactly with the 
# theoretical quantities




############################## Simulating the geometric random variable and its expectation

p <- .2  # choose the probability of success

n <- 10000
n.trials <- rep(0,n) # create a vector to store the simulation results

for (i in 1:n) {
	stop.trials <- F
	counter <- 0
	
	while (stop.trials == F){
		r <- runif(1,0,1)      # generate 1 random number between 0 and 1
		counter <- counter + 1
		if (r < p) {
			n.trials[i] <- counter 
			stop.trials <- T
		}
	}	
}
n.trials

s <- seq(from=0.5,by=1,to=max(n.trials)+.5)
hist(n.trials,breaks=s,freq=F) # better histogr
c(1/p,mean(n.trials))