MATH103

Definition: Sample Space (Ω)

The set of all possible outcomes of a random experiment.

Definition: Event

A subset of the sample space Ω, representing a collection of outcomes.

Definition: Discrete Uniform Law

If all outcomes in a finite sample space Ω are equally likely, the probability of an event A is P(A) = |A| / |Ω|.

Axiom 1 of Probability (Non-negativity)

For any event A, P(A) ≥ 0.

Axiom 2 of Probability (Normalization)

The probability of the entire sample space is 1, i.e., P(Ω) = 1.

Axiom 3 of Probability (Countable Additivity)

If A1, A2, A3, … are pairwise disjoint events, then P(∪ A_i) = ∑ P(A_i).

Property: Complement Rule

P(Aᶜ) = 1 - P(A)

Property: Inclusion-Exclusion Principle (2 events)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Definition: Conditional Probability

The probability of event A occurring given that event B has occurred: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

Theorem: Law of Total Probability

If B1, B2, …, Bn partition the sample space Ω, then for any event A: P(A) = ∑ P(A | B_i) * P(B_i).

Theorem: Bayes' Theorem

P(B_i | A) = [P(A | B_i) * P(B_i)] / P(A), used to update the prior probability of an event after observing evidence.

Definition: Independent Events

Two events A and B are independent if and only if P(A ∩ B) = P(A) * P(B).

Definition: Discrete Random Variable

A variable that takes a countable number of distinct values, each with a specific probability.

Definition: Probability Mass Function (p.m.f.)

A function p_X(x) that gives the probability that a discrete random variable X is exactly equal to x: p_X(x) = P(X = x).

Definition: Expected Value (Discrete)

The long-run average or mean value of a discrete random variable: E[X] = ∑ x * p_X(x).

Definition: Variance (Var(X))

A measure of the spread of a random variable around its mean: Var(X) = E[(X - E[X])²] = E[X²] - (E[X])².

Model: Bernoulli Distribution

A discrete distribution modeling a single trial with two outcomes (Success = 1, Failure = 0), where P(X=1) = p.

Model: Binomial Distribution (X ~ Bin(n, p))

Models the number of successes in n independent Bernoulli trials, with p.m.f. P(X = k) = nCk * p^k * (1-p)^(n-k).

Model: Geometric Distribution

Models the number of independent Bernoulli trials needed to get the first success; P(X = k) = (1-p)^(k-1) * p.

Model: Poisson Distribution (X ~ Po(λ))

Models the number of events occurring in a fixed interval of time or space; p.m.f. P(X = k) = (e^-λ * λ^k) / k!.

Definition: Continuous Random Variable

A random variable whose possible values form an uncountable set (usually an interval of real numbers).

Definition: Probability Density Function (p.d.f.)

A integrable function f_X(x) used to find probabilities for a continuous random variable: P(a ≤ X ≤ b) = ∫_{a}^{b} f_X(x) dx.

Definition: Cumulative Distribution Function (c.d.f.)

The function F_X(x) expressing the probability that X takes a value less than or equal to x: F_X(x) = P(X ≤ x).

Model: Uniform Distribution (X ~ U(a, b))

A continuous distribution where all intervals of equal length in the support [a, b] are equally likely.

Model: Exponential Distribution

A continuous distribution often used to model the time between independent Poisson events, with p.d.f. f(x) = λe^{-λx} for x ≥ 0.

Model: Normal Distribution (Gaussian)

A symmetric, bell-shaped continuous distribution parameterized by its mean (μ) and variance (σ²).