Tutorial 01

Problem 01

In R, and using just one line of code, create the following matrices and assign them to the variables W, X, Y, and Z, respectively. $$\begin{align} \texttt{W} = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix},~ \texttt{X} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix},~ \texttt{Y} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \text{ and } \texttt{Z} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}. \end{align}$$ Also, compute $\texttt{W}\odot\texttt{X}$ and $\texttt{Y} \cdot \texttt{Z}$, such that $\odot$ represents the element-wise multiplication and $\cdot$ matrix multiplication.

Problem 02

Create a matrix $[\texttt{A}]_{3 \times 3}$, such that $\texttt{a}_{ij} = i + j$, $\forall i, j$. Also, create $[\texttt{B}]_{3 \times 3}$, such that $\texttt{b}_{ij} = \texttt{TRUE}$, if $i + j$ is even, and $\texttt{b}_{ij} = \texttt{FALSE}$, otherwise. Finally, create $[\texttt{C}]_{3 \times 3}$, such that $\texttt{c}_{ij} = 1$, if $i + j \in \{1, 2, 3\}$, and $\texttt{c}_{ij} = 0$, otherwise.

Problem 03

Write a function for computing $n!$, for any $n$ non-negative integer. Print the 10 first numbers from the sequence $(\texttt{f}_n)_n$, such that $\texttt{f}_n = n!$, for all $n \in \mathbb{Z}_{+}$. Also, for $n = 10$, compare your result with the function from (recall that, for any positive integer $n$, $\Gamma(n) = (n - 1)!$).

Problem 04

Create 3 five-element (row) vectors $\texttt{p}$, $\texttt{q}$, and $\texttt{r}$, such that $p$ contains the first 5 prime numbers, $q$ the first 5 numbers from the Fibonacci sequence, and $r$ the 5 first elements of the sequence $(\texttt{r}_n)_n$, such that $\texttt{r}_n = \lfloor\sqrt{(n \cdot \pi)}\rfloor$, $\forall n$, and $\lfloor \cdot \rfloor$ is the floor function. Do NOT compute the terms manually (write functions!), so that you can easily extend it to larger vectors. Also, create the matrices $$\begin{align} \texttt{A} = \begin{pmatrix} \texttt{p} \\ \texttt{q} \\ \texttt{r} \end{pmatrix}, \text{ and } \texttt{B} = \begin{pmatrix} \texttt{p}^{\text{T}} & \texttt{q}^{\text{T}} & \texttt{r}^{\text{T}} \end{pmatrix}. \end{align}$$

Problem 05

Suppose that, for the population of male teenagers, we can model their height as $$\begin{align} \texttt{height}_i = \beta_0 + \beta_1 \cdot \texttt{age}_i + \epsilon_i, \text{ such that } \epsilon_i \sim \text{Normal}(0, \sigma^2_{\epsilon}). \end{align}$$ Now, for a random sample of size $n = 20$, we want to estimate $\boldsymbol{\beta} = (\beta_0, \beta_1)^{\text{T}}$ as $\hat{\boldsymbol{\beta}} = (\text{X}^{\text{T}}\text{X})^{-1}\text{X}^{\text{T}}\boldsymbol{y}$, such that $$\begin{align} \text{X} = \begin{pmatrix} 1 & \texttt{age}_1 \\ \vdots & \vdots \\ 1 & \texttt{age}_{20} \end{pmatrix}, \text{ and } \boldsymbol{y} = \begin{pmatrix} \texttt{height}_1 \\ \vdots \\ \texttt{height}_{20} \end{pmatrix}. \end{align}$$ Compute $\hat{\boldsymbol{\beta}}$ and plot $\hat{\boldsymbol{y}} = \text{X}\hat{\boldsymbol{\beta}}$. For this exercise, generate the data set using the following code

set.seed(999)
beta_0 <- 140
beta_1 <- 2
ages <- sample(x = 12:18, size = 20, replace = TRUE)
heights <- round(x = beta_0 + beta_1 * ages + rnorm(n = 10, mean = 0, sd = 2), 
                 digits = 2)
data <- data.frame(heights = heights, ages = ages)