### Simulate flipping a fair coin.
### We will use the Uniform Distribution

# simulate one flip of a fair coin

p <- 0.5        # assume a fair coin

u <- runif(1)   # gerneate 1 uniform random number
u               # print the value

h <- 0              # assume h is zero, i.e., flip is a T
if(u < p) h <- 1    # if u is < p consider the value a H
                    # if u is > p consider the value a T
h                   # print the simulated value

# simulate m flips of a fair coin

m <- 1000   # total number of times the coin is flipped,
            # i.e., the total number of times the experiment is performed

h <- numeric(m)     # initialize a vector for H
h                   # prints the vector of zeros

for(i in 1:m){              # a loop that simulates m flips
    u <- runif(1)
    if(u < p) h[i] <- 1
}

h           # print out the results

h[1:10]     # print out the first 10 values of h


sum(h)      # the total number of h's in the m flips

p.hat.h <- sum(h)/m    # the estimated probability of h's
p.hat.h

i <- seq(1,m)       # generate a sequence from 1 to m

h.c <- cumsum(h)    # computes the cummulate sum of values

h.hat <- h.c/i      # relative frequency by the index
h.hat

plot(h.hat)