Understanding uncertainty, data analysis, and statistical inference
← Back to MathematicsLearn the basic concepts and axioms of probability theory.
Study discrete and continuous random variables and their distributions.
Master the most important discrete and continuous probability distributions.
Study multiple random variables and their relationships.
Learn the fundamental limit theorems that justify statistical methods.
Learn to summarize and visualize data effectively.
Understand how sample statistics relate to population parameters.
Learn methods for estimating population parameters from sample data.
Master the logic and methods of statistical hypothesis testing.
Learn to compare means across multiple groups and factors.
Study relationships between variables using regression methods.
Explore modern statistical methods and their applications.
Build the mathematical foundation for probability theory and learn the basic rules of probability.
Learn sample space Ω as the set of all possible outcomes. Define events as subsets of the sample space and study set operations on events.
Master Kolmogorov's axioms: P(Ω) = 1, P(A) ≥ 0 for all events A, and countable additivity for disjoint events. Derive basic probability rules.
Define P(A|B) = P(A ∩ B)/P(B) and understand it as probability of A given information about B. Learn multiplication rule and its applications.
Learn that events A and B are independent if P(A ∩ B) = P(A)P(B). Understand independence vs. disjointness and mutual independence.
For partition {B₁, B₂, ...}, learn P(A) = Σ P(A|Bᵢ)P(Bᵢ). Use this law to break complex problems into simpler conditional problems.
Learn P(Bᵢ|A) = P(A|Bᵢ)P(Bᵢ)/P(A) for updating probabilities with new information. Apply to medical diagnosis, spam filtering, and other inverse problems.
Master fundamental counting principle, addition principle, and inclusion-exclusion. Learn to count outcomes systematically for probability calculations.
Learn P(n,r) = n!/(n-r)! for arrangements and C(n,r) = n!/(r!(n-r)!) for selections. Apply to probability problems involving equal likelihood.
Learn how to model uncertain quantities using random variables and their distributions.
Define random variable X as a function from sample space to real numbers: X: Ω → ℝ. Understand how this maps outcomes to numerical values.
Study random variables with countable range. Learn that all probability mass concentrates at specific values with positive probability.
Define PMF as p(x) = P(X = x) for discrete X. Learn that PMF satisfies p(x) ≥ 0 and Σp(x) = 1 over all possible values.
Study random variables with uncountable range. Learn that P(X = x) = 0 for any specific value, so we work with intervals instead.
Define PDF f(x) where P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx. Learn that f(x) ≥ 0 and ∫₋∞^∞ f(x)dx = 1.
Define CDF as F(x) = P(X ≤ x) for all real x. Learn properties: F is non-decreasing, right-continuous, with F(-∞) = 0 and F(∞) = 1.
Learn E[X] = Σx·p(x) or ∫x·f(x)dx as the "center" of distribution. Define Var(X) = E[(X - μ)²] as measure of spread.
Define MGF as M(t) = E[e^(tX)]. Learn how MGFs uniquely determine distributions and simplify calculations involving sums of random variables.
Master the most important probability distributions and their applications.
Study Bernoulli(p) for single success/failure trial and Binomial(n,p) for number of successes in n independent trials. Learn formulas and applications.
Learn Geometric(p) for number of trials until first success and NegativeBinomial(r,p) for trials until r-th success. Understand memoryless property.
Study Poisson(λ) for counting rare events in fixed time/space. Learn as limit of binomial when n → ∞, p → 0, np → λ. Apply to queuing, reliability.
Learn Uniform(a,b) with constant density on [a,b]. Understand as model for "equally likely" outcomes and its use in simulation and random number generation.
Master Normal(μ,σ²) as the most important continuous distribution. Learn standard normal, 68-95-99.7 rule, and central role in statistics via CLT.
Study Exponential(λ) for modeling waiting times. Learn memoryless property and relationship to Poisson process. Apply to reliability and survival analysis.
Learn these distributions from normal: χ² for sums of squared normals, t for ratios involving sample variance, F for ratios of χ² variables. Essential for inference.
Study Gamma(α,β) as generalization of exponential and Beta(α,β) for probabilities on [0,1]. Learn their flexibility for modeling various phenomena.
Study multiple random variables simultaneously and their relationships.
Define joint PMF p(x,y) = P(X=x, Y=y) for discrete variables and joint PDF f(x,y) for continuous variables. Learn to compute probabilities over regions.
Obtain marginal distributions by "summing out" or "integrating out" other variables: pₓ(x) = Σᵧ p(x,y) or fₓ(x) = ∫ f(x,y)dy.