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