What you will learn

• Probability
• Distributions
• Hypothesis testing
• Regression
• Statistics

Course content (11 lessons)

1. Unit I: Introduction to Probability and Rules — Unit I: Random Variables and Distribution Functions 1. Introduction to Probability Definitions: Experiment: A process leading to an outcome (e.g., rolling a die). Sample Space…
2. Joint Probability and Random Variables — 1.2 Joint Probability Definition: Probability of two specific events happening together. Example (Balls in a Bag): A bag has 4 Red Balls (R) and 6 Blue Balls (B). Two balls are…
3. Distribution Functions: PMF, PDF, CDF — 3. Distribution Functions 3.1 Probability Mass Function (PMF) For: Discrete Variables. A table or function listing probabilities for specific values. Example (Rolling a Die): Let…
4. Discrete Probability Distributions — 4. Discrete Probability Distributions 1. Binomial Distribution Use: Fixed number of trials ($n$), two outcomes (Success/Failure), independent trials. Example (Quality Control): A…
5. Continuous Probability Distributions — 5. Continuous Probability Distributions 1. Uniform Distribution Use: Every value in a range is equally likely. Example (Random Number Generator): A computer generates a random…
6. Expectation and Variance — Unit II: Moments and Moment Generating Functions 1. Expectation of a Random Variable Concept The Expected Value $E[X]$, often denoted by $\mu$, represents the theoretical average…
7. Moments: Raw and Central — 2. Moments Moments are quantitative measures that describe the shape of a probability distribution. 2.1 Raw Moments ($r^{th}$ Moment about Origin) Denoted by $\mu' r$. Definition:…
8. Probability Generating Function (PGF) — 3. Probability Generating Function (PGF) Concept The PGF is a power series representation of the probability distribution of a discrete random variable. It converts a probability…
9. Moment Generating Function (MGF) — 4. Moment Generating Function (MGF) Concept Similar to PGF, but works for both discrete and continuous variables. It uses the exponential function $e^{tX}$. Definition $$ M X(t) =…
10. Two-Dimensional Random Variables — 5. Two-Dimensional Random Variables When dealing with two variables $X$ and $Y$ simultaneously (e.g., Height and Weight). 5.1 Joint Distribution Function Definition: $F(x, y) =…
11. Independence and 2D Example — 5.4 Independence of Random Variables Concept: $X$ and $Y$ are independent if the occurrence of one does not affect the probabilities of the other. Mathematical Condition: $X$ and…

Unit I: Introduction to Probability and Rules

Unit I: Random Variables and Distribution Functions

1. Introduction to Probability

Definitions:

• Experiment: A process leading to an outcome (e.g., rolling a die).
• Sample Space ($S$): Set of all outcomes (e.g., ${1, 2, 3, 4, 5, 6}$).
• Event ($E$): A subset of outcomes (e.g., Rolling an even number ${2, 4, 6}$).

1.1 Probability Rules

1. Addition Rule (Sum of Probability): Used for "OR" situations ($A \cup B$).

• Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Example (Drawing a Card):
• Let $A$ be drawing a King (4 cards).
• Let $B$ be drawing a Heart (13 cards).
• There is 1 card that is both (King of Hearts), so $P(A \cap B) = 1/52$.
• Probability of drawing a King OR a Heart:

$$ P(A \cup B) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} $$

2. Multiplication Rule (Product of Probability): Used for "AND" situations ($A \cap B$).

• Formula: $P(A \cap B) = P(A) \times P(B|A)$.
• Example (Drawing Two Aces):
• You draw two cards from a deck without replacement.
• Event $A$: First card is an Ace ($P(A) = 4/52$).
• Event $B$: Second card is an Ace. Since one Ace is gone, $P(B|A) = 3/51$.
• Probability of Two Aces:

$$ P(A \cap B) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} $$

Joint Probability and Random Variables

1.2 Joint Probability

• Definition: Probability of two specific events happening together.
• Example (Balls in a Bag):
• A bag has 4 Red Balls (R) and 6 Blue Balls (B). Two balls are drawn.
• Joint Probability of drawing Red first AND Blue second (without replacement):

$$ P(R \cap B) = P(R) \times P(B|R) = \frac{4}{10} \times \frac{6}{9} = \frac{24}{90} = 0.266 $$

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2. Random Variables

A variable whose value depends on the outcome of a random experiment.

2.1 Types of Random Variables

1. Discrete Random Variable:

• Takes specific, countable values.
• Example: You flip a coin 3 times. Let $X$ = Number of Heads.
• Possible values for $X$: ${0, 1, 2, 3}$. (You cannot get 2.5 heads).

2. Continuous Random Variable:

• Takes any value within a range (infinite possibilities).
• Example: Let $Y$ = Height of a student in a class.
• Possible values: Could be 150.1 cm, 150.12 cm, 160.5 cm, etc. It is measured, not counted.

Frequently asked questions

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