RStudio
The default layout is slightly different, and the default theme is light. Both can be changed in the settings.
A typical RStudio window looks like this:
- In the Console, we can execute R commands. For example, we can type
1+1and press Enter to confirm. - The Environment shows all currently defined variables and datasets.
- In the Plots pane, generated graphics are displayed.
- In the Help pane, we can access the documentation.
- In the large pane on the left, we can open a script.
Script vs. Console
- With a script, we can save different lines of code and execute them whenever needed.
- We can also execute code directly in the console, but we cannot save it there.
- In the script, we can place the cursor on a line and execute it with Ctrl+Enter (Linux, Windows) or Cmd+Enter (macOS). It will then be “inserted” into the console.
- Alternatively, we can highlight multiple lines and execute them, or execute the entire script at once.
Dimensions and Transposition
On the previous slide, \(\boldsymbol{X}\) had dimensions \(n\times k\), and \(\boldsymbol{x}\) had dimension \(n\). \(\boldsymbol{x}\) was written as a column vector, but we can also write \(\boldsymbol{x}\) as a row vector. To do this, we must transpose the vector: Simply put, rows become columns, and columns become rows. We denote the transposition with a small apostrophe (or optionally a superscript T):
\[ \boldsymbol{x}' = (x_1, x_2, \dots, x_n) \]
We can also transpose matrices. A matrix that previously had dimensions \(n\times k\) will have dimensions \(k\times n\) after transposition.
If we transpose a transposed matrix, we get:
\[ (\boldsymbol{X}')' = \boldsymbol{X} \]
Furthermore, \((\boldsymbol{XZ})'=\textcolor{red}{\boldsymbol{Z}'\boldsymbol{X}'}\).
Discrete Random Variables
A discrete random variable is a random variable that can have only a finite or countably infinite number of possible outcomes. If the variable is called \(X\), we denote the outcomes as \(x_i\) and the corresponding probabilities as \(p_i\). Note that the sum of all probabilities \(\sum_i p_i\) must equal 1.
An example of a discrete random variable would be the roll of two dice. The possible outcomes are \(\{2,3,4,5,6,7,8,9,10,11,12\}\), and the corresponding probabilities are \(\{\tfrac{1}{36},\tfrac{2}{36},\tfrac{3}{36},\tfrac{4}{36},\tfrac{5}{36},\tfrac{6}{36},\tfrac{5}{36},\tfrac{4}{36},\tfrac{3}{36},\tfrac{2}{36},\tfrac{1}{36}\}\). To the right, the probability mass function (PMF) is illustrated:
A Bernoulli variable is a discrete random variable that can only take on two outcomes, such as a coin toss.
Continuous Random Variables (2)
An example of such a variable would be the height of a person. It would make no sense to ask for the probability that a person is exactly 1.734681092536 meters tall. This probability is zero. But we can look at the PDF and determine how likely it is that the person’s height lies between 1.73 and 1.74 meters:
Whether an infinite set of numbers is countable or uncountable can be intuitively answered. All natural numbers \(\mathbb{N}\) are countably infinite. We can clearly specify a way to count them (start at 0, then 1, then 2, then 3, …), we just don’t know where and when the path ends. After all, it is still infinitely long. All real numbers \(\mathbb{R}\), however, are uncountably infinite. We cannot define a unique way to include all numbers. Suppose we start at 0, then 0.001 what about all the numbers in between? And all the numbers between those numbers? There is no way to count them all.
Cumulative Distribution Function
In addition to the probability density function, we can draw the cumulative distribution function (CDF). It represents the probability that the outcome is less than or equal to a certain value. The function is strictly monotonically increasing:
The dashed line shows how to read the plot: The value of the density function at \(X=1.74\) represents the probability that a randomly selected person is shorter than or exactly 1.74 meters tall.
