R basics.

Please, create functions for each of the tasks. Functions shouls be named as task names, usually, as task1, task2, etc. Here are some function examples:

fun1 <- function(x, y) { # here are arguments in the brackets
  x + y # the result of the function is its last expression
}

# let's now call the function
fun1(10, 20) # this is 30

# another example
fun2 <- function(n) {
   c(1:n, n:1)
}

# let's test the function
fun2(5) # this is a vector 1, 2, 3, 4, 5, 5, 4, 3, 2, 1

fun3 <- function() {
    x <- rbinom(20, size=1, prob=0.5)  # toss a coin 20 times
    mean(x) # this is the result of a function, because this is the last expression
}

fun3() # the result is always different, but it should be about 0.5

1. task1 Read help about the rep function (type help(rep)). Then use it to produce a vector of repeating numbers 1, 2, and 3: \(1, 2, 3, 1, 2, 3, 1, 2, 3, \ldots\). The length of the vector should be 40.
   • task1a You are given integers n and size, create a vector of repeating numbers from 1 to n of size size, for example, task1a(3, 10) should return c(1, 2, 3, 1, 2, 3, 1, 2, 3, 1).
2. task2 You are given integers n and size. Using the function sample (read help if needed) generate a vector of size size consisting of random integers from 1 to n, and return it. For example, task2(3, 5) may return 1, 3, 2, 3, 3.
   • task2a call task2a with the arguments n=5 and size=100, then use functions table to find out, how many times each number was generated, use print to print the table from inside the function.
3. filter.k You are given a vector x and a number k. Return a new vector, that is a copy of x but without elements equal to k. For example, filter.k(c(1, 2, 3, 4, 3, 2, 1), 3) should return 1, 2, 4, 2, 1.
4. random.walk.1d You are given an integer steps, generate a random vector of size steps consisting of numbers 1 and -1. Return the sum of its elements.
5. mixed.distribution. You are given an integer size. Generate a vector of size size using the following algorithm: to generate the next number, toss a coin (50%/50% of head and tails). If you get heads, generate a number from distribution\(N(0, 1)\). Otherwise, from a uniform distribution with min=-1 and max=1.
6. random.walk.2d Random walk on a plane. You are given an integer steps. Generate steps times a pair of numbers, this pair may be either (1 0), (-1 0), (0 1), or (0 -1). Put all this pairs in one vector, so you get a vector of size 2 * size. This pairs correspond to movements of a point on a plane, a pair is a change for x and y coordinates correspondingly. That is a point moves either up, left, right or down. Finally, sum all x coordinates (all odd indexes), then sum all y coordinates (all even indexes), and you will get two coordinates of where a point had come after its random walk.
   • plot.random.walk You are given interers n and size. Call the previous function with the size argument n times. Plot all the n points.
7. kinder.surprise There are n possible toys inside a kinder surprise. How many kinder surprises should one buy in avarage to get all the toys? Make 10000 experiments, where you repeat opening kinder surprises untill you get all the toys. Then compute an avarage number of steps you needed to finish.