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:

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