# Week 1 Lecture 2

## Frequency and Relative Frequency
In a dataset for some **discrete** (=finite outcomes) variable x:
* The **frequency** of any particular value that x can take is defined to be the number of occurrences of that value in the dataset.
* The **relative frequency** of a value is defined to be the fraction or proportion of occurrences of that value in the dataset:
$$
\begin{array}{c}
\text { relative frequency } \\
\text { of a value } k
\end{array}=\frac{\text { number of times the value } k \text { occurs }}{\text { total number of observations in the data set }}
$$
### Examples

## Histograms
Steps to construct a histogram:
1. Choose the number of classes for variable x. A general rule of thumb is:
$$
\text{number of classes}
\approx
\sqrt{\text{number of observations}}
$$
2. Determine the frequency and relative frequency of each possible class fo x.
3. Mark all possible classes on the horizontal axis.
4. Draw a rectangle/bar whose height is the **frequency** or **relative frequency** of the corresponding class.

### Examples

### Bivariate histograms
Histograms that show the distribution of two variables.

## Set Theory
### Operations in Set Theory
* **Complement** of an event A, A^c^, is the set of all outcomes in Ω (sample space) that are not contained in A. i.e. Ω\A
* **Intersection** of two events A and B, is denoted by A ∩ B is the event consisting of all outcomes that are in **both** A and B.
* **Union** of two events A and B, A ∪ B, is the event consisting of all outcomes that are **either** in A or in B.


### Events
**Null event** is an event that consists of no outcomes, ∅. This symbol ∅ is also called the "empty set" or "null set";.

Events A and B are said to be **mutually exclusive** or **disjoint** events if A ∩ B = ∅.

Events A~1~, A~2~, A~3~,... are mutually exclusive (or pairwise disjoint) if no two events have any outcome in common.

### DeMorgan's Laws
$$
(A \cup B)^{c}=A^{c} \cap B^{c} \\
(A \cap B)^{c}=A^{c} \cup B^{c}
$$

## Probability
Given an experiment and a sample space Ω, the **probability** of an event A is defined to be a number that represents some measure of the chance that A will occur. This is denoted as Pr(A) (or also P(A)).

Let X denote the outcome variable from tossing a coin, and H represents the outcome of tossing heads.
We can write Pr({H})=0.5 or Pr(X=H) = 0.5, to mean that event {H} has a probability of 0.5. {H} means a compound event of H.

### Axioms of Probability
* Axiom 1: For any event A, $Pr(A) \ge 0$
* Axiom 2: $Pr(\Omega) = 1$
* Axiom 3: Any (countably) infinite sequence of mutually exclusive (disjoint) events A~1~, A~2~, A~3~,... satsifies:
$$
\operatorname{Pr}\left(A_{1} \cup A_{2} \cup A_{3} \cup \ldots\right)=\sum_{i=1}^{\infty} \operatorname{Pr}\left(A_{i}\right)
$$
#### Example

**Proof that null events have zero probability**

**Proof regarding finite sequence of mutually exclusive events**

**Proof regarding complements of events**

**Proof regarding range of probability**

### Properties of Probability
* Pr(∅) = 0
* For mutually exclusive events A~1~,...,A~n~,
$Pr(A_1\cup...\cup A_n) = Pr(A_1)+...+Pr(A_n)$
* For any event A, $Pr(A^c)=1-Pr(A)$
* For any event A, $0 \le Pr(A) \le 1$.

### ∅
For a null event ∅, we know that Pr(empty;) = 0.
However, Pr(A) = 0 does not mean we must have A=∅
e.g.
Randomly pick a number n in the range [1,∞). The probability that n=500 is 0, but the event {n=500} is not a null event.

### 1
Probability 1 does not mean "always occurs".
Conversely, using the same example, P(n≠500) =1. However, this does not mean that P is always not equals to 500.

### Example