Unit 3: Probability Component

PROBABILITY

• The probability model
• Sample space
• Events
The class of events includes the sample space and is closed under complements and countable unions and intersections.
• Probability
Postulates:
• Nonnegativity
• Normalization
• Additivity and countable additivity
• Rules for computing probabilities
• Discrete probability models: the sample space is countable.
• Equally likely outcomes: P(A) = #A/#S. This relates probability & combinatorics.
• Conditional probability
• Definition
• Computing P(A | B) by reducing the sample space to B
• For fixed B, P(A | B) as a function of event A follows the rules of probability.
• Multiplication rules for probabilities of intersections
• Independent events
• Law of Total Probability and Bayes's rule
• Random variables (r.v.s) and their distributions
• cdf: the (cumulative) distribution function
• A discrete r.v. X has a countable range. Its distribution is determined by specifying P(X=x) for each possible x (called the probability [mass] function or discrete density).
• A continuous r.v. is one whose cdf is continuous. If that cdf has a derivative (function) f, then f is the (probability) density (function) (pdf) of the r.v.
• Bernoulli trials
• Special discrete probability distributions (as many as we cover before the exam)
• Uniform distributions
• Binomial distributions B(n,p); B(1,p) = Bernoulli distribution.
• Geometric distributions
• Negative binomial distributions
• Hypergeometric distributions
• Poisson distributions