Fundamentals of Probability and Distributions

📋 Course Outline

1. Probability Foundations
2. Sample Space and Events
3. Conditional Probability
4. Independence
5. Random Variables
6. Discrete Distributions
7. Continuous Distributions
8. Expectation and Variance
9. Central Limit Theorem
10. Distribution Applications

📖 1. Probability Foundations

🔑 Key Concepts & Definitions

• Probability (P): A numerical measure between 0 and 1 that quantifies the likelihood of an event occurring; 0 indicates impossibility, 1 indicates certainty.

• Experiment: A procedure or process that results in one outcome from a set of possible outcomes, used to observe random phenomena.

• Sample Space (S): The complete set of all possible outcomes of an experiment; denoted as S.

• Event: A subset of the sample space, representing one or more outcomes; can be simple (single outcome) or compound (multiple outcomes).

• Conditional Probability (P(A|B)): The probability that event A occurs given that event B has already occurred, calculated as ( P(A|B) = \frac{P(A \cap B)}{P(B)} ).

• Independence: Two events A and B are independent if the occurrence of one does not affect the probability of the other, i.e., ( P(A \cap B) = P(A) \times P(B) ).

📝 Essential Points

• Probabilities are assigned based on the ratio of favorable outcomes to total outcomes in equally likely scenarios.