# Week 2

## Learning Outcomes

### First Lecture

Define sample space and events

Define probability and rules

Define random variable, probability mass function, and cumulative density function

### Second Lecture

Describe disjoint events

Describe a conditional probability

Define independent events

Law of Total Probability

Baye’s Theorem

## Important Concepts

### First Lecture

#### Experiment

A process that yields an outcome randomly.

#### Sample Space

The *sample space* (S) is all the possible outcomes from an experiment.

#### Event

An *event* is a collection of outcomes of an experiment that is any subset of the sample space S (including S).

#### Random Variable

A *random variable* (rv) is a function that maps a sample space to the real numbers. A rv is denoted by a capital letter. The observed value is denoted with a lower-case letter.

### Probability (Cumulative) Distribution Function

The *cumulative distribution function* of a random variable \(X\), expressed as \(F_X(x)\), is defined as

\[ F_X(x) = P(X\leq x), \]

providing the probability of observing \(x\) or lower.

#### Probability Mass Function (Categorical Variable^{1})

The *probability mass function* of a random variable, expressed as \(f_X(x)\), is expressed as

\[ f_X(x)=P(X=x), \]

providing the probability of observing \(x\).

### Second Lecture

**Disjoint Events**

Two events **A** and **B** are considered disjoint if \(P(A\cap B)=0\). In general terms, only one event can occur, not both.

#### Conditional Probability

Let there be 2 events **A** and **B**. Given that **B** has occurred, what is the probability that **A** occurs? The conditional probability requires there to be at least 2 events and one event must have occurred. Additionally, the events cannot be disjoint. Conditional probabilities are denoted as $P(A|B)$, the probability of **A** given **B** has occurred.

- \(P(A|B)=\frac{P(A\cap B)}{P(B)}\)

#### Independent Events

Events **A** and **B** are considered independent if \(P(A\cap B)=P(A)P(B)\). In other words, The occurrence of one event will not affect the occurrence of the other event.

#### Law of Total Probability

The law of total probability allows you to compute the probability of an event **A** given that the sets {\(B_1\), …, \(B_n\)} partitions event **A**. The law of total probability is given as

\[ P(A) = \sum^n_{i=1}P(A\cap B_i) = \sum^n_{i=1}P(A|B_i)P(B_i) \]

#### Baye’s Theorem

Baye’s theorem computes the probability of an event \(B_i\) given event **A**

\[ P(B_i|A) = \frac{P(A\cap B_i)}{P(A)}=\frac{P(A|B_i)P(B_i)}{\sum^n_{i=1}P(A|B_i)P(B_i)} \]

## Resources

### First Lecture

Slides | Videos |
---|---|

Slides | Video 001 Video 002 |

### Second Lecture

Slides | Videos |
---|---|

Slides | Video 001 Video 002 |

## Footnotes

We will define this later on in the semester.↩︎