# Week 2

This week, we will briefly discuss topics related to conditional probabilities. Additionally, we will introduce commonly used distribution functions.
Published

January 30, 2023

## 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 Variable1)

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

1. We will define this later on in the semester.↩︎