Unit 3: Probability Component
- The probability model
- Sample space
The class of events includes the sample space and
is closed under complements and countable unions and intersections.
- 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
- 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
- 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