################
### EXERCISE 1.1
################
### Simulation of flipping a coin.
### Change the probabilities for "head" and "tail",
### change the number of flips (argument size),
### and study the distribution of the outcome
### SEE LECTURE: Definitions 1 & 3
S = sample(c("head", "tail"), size=10, replace=TRUE, prob=c(0.5, 0.5))
S
table(S)
prop.table(table(S))




################
### EXERCISE 1.2
################
### Simulation of tossing the dice.
### Change the number of tosses (argument "size"),
### and study the distribution of the outcome
### SEE LECTURE: Definition 2 & 3
S = sample(1:6, size=10, replace=TRUE)
S
table(S)
prop.table(table(S))

### Use the operators ">", ">=", "<", "<==", and "=="
### to obtain frequencies for specified subsets
S>2
table(S>2)
prop.table(table(S>2))



################
### EXERCISE 1.3
################
### Calculate the number of possible combinations 
### when drawing k elements out of n, using the binomial coefficient.
### Use different values of k and n.
### SEE LECTURE: Theorem 1
n = 3
k = 2
### Combinations without replicates
choose(n, k)
### Combinations with replicates
choose(n+k-1, k)

### Calculate all possible combinations (without replicates)
### when taking k elements out of the set x.
### Choose other sets x (e.g. weekdays) and other values of k.
x = c("apple", "banana", "orange")
k = 2
combn(x, k)
### Alternative with package "gtools":
library(gtools)
combinations(3, 2, x, repeats=FALSE)

### Calculate all possible combinations (with replicates)
### when taking k elements out of the set x.
### Choose other sets x (e.g. weekdays) and other values of k.
library(gtools)
combinations(3, 2, x, repeats=TRUE)



################
### EXERCISE 1.4
################
### Calculate the number of possible variations
### when drawing k elements out of n.
### Use different values of k and n.
### SEE LECTURE: Theorem 2
n = 3
k = 2
### Variation without replicates
y = n:(n-k+1)
y
prod(y)
### Variation with replicates
n^k

### Calculate all possible variations (without replicates)
### when taking k elements out of the set x.
### Choose other sets x (e.g. weekdays) and other values of k.
x = c("apple", "banana", "orange")
k = 2
permutations(3, 2, x, repeats=FALSE)

### Calculate all posibble variations (with replicates)
### when taking k elements out of the set x.
### Choose other sets x (e.g. weekdays) and other values of k.
permutations(3, 2, x, repeats=TRUE)
### Alternative way:
expand.grid(x, x)



################
### EXERCISE 1.5
################
### Let X be the number of food products that can be produced by a company,
### and let P be the probability that the components for each number of
### products are available on the market.
### SEE LECTURE: Example 4
X = c(0, 1, 2, 3, 4)
P = c(0.2, 0.3, 0.2, 0.2, 0.1)

### Calculate the expectation of X, i.e. the average expected number
### of products that can be produced.
### Calculate the standard deviation of X, i.e. the average deviation from
### the expectation.
### SEE LECTURE: Definitions 8 and 9
sum(X * P)
sqrt(sum(X^2 * P) - sum(X * P))

### Draw the distribution of X using a bar chart.
### Change arguments, col, lwd, cex.axis, cex.lab, xlab, ylab, main, cex.main
plot(X, P, type="h", col="blue", lwd=3, cex.axis=1.5, cex.lab=1.5, xlab="Number X of products", ylab="P(X)", main="Distribution", cex.main=1.5)

### Draw the cumulative distribution function of X.
### Modify graphic parameters.
### What is the probability that 3 or less products can be produced.
cumsum(P)
SF = stepfun(X, c(0, cumsum(P)))
plot(SF, col="grey", lwd=1, lty=3, cex.axis=1.5, cex.lab=1.5, xlab="Number X of products", ylab="F(X)=P(x<=X)", main="Cumulative distribution function", cex.main=1.5, verticals=TRUE, do.points=FALSE)
lines(SF, col="blue", lwd=3, verticals=FALSE, do.points=FALSE)