---
title: "Lecture 1: Exercises"
date: September 27th, 2018
output: 
  html_notebook:
    toc: true
    toc_float: true
---

# Exercise 1: Vectors

1. Generate and print a vector of 10 random numbers between 5 and 500.
2. Generate a random vector Z of 1000 letters (from "a" to "z"). Hint: the 
variable `letters` is already defined in R.
3. Print a summary of Z in the form of a frequency table.
4. Print the list of letters that appear an even number of times in Z.

# Exercise 2: Matrices

1. Create the following 5 by 5 matrix and store it as variable X.
```{r echo = FALSE}
matrix(1:30, nrow = 5, byrow = FALSE)
```

2. Create a matrix Y by adding an independent Gaussian noise (random numbers)
with mean 0 and standard deviation 1 to each entry of X. e.g.

3. Find the inverse of Y.

4. Show numerically that the matrix product of Y and its inverse 
is the identity matrix.


# Exercise 3: Data fames

1. Create the following data frame and name it “exams”.

```{r, echo = FALSE}
set.seed(123)
data.frame(
  student = c("Alice", "Sarah", "Harry", "Ron", "Kate"),
  score = sample(80:100, 5),
  letter = sample(c("A","B"), 5, replace = TRUE),
  late = sample(c(T, F), 5, replace = TRUE)
)
```

2. Compute the mean score for this exam and print it.
3. Find the student with the highest score and print the corresponding 
row of "exams". Hint: use the function `which.max()`.