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- 1. Stat310 Probability and Statistics Hadley Wickham
- 2. 1. Two important facts 2. Syllabus 3. Introduction to probability 4. Deﬁnitions & properties 5. Probability as a set function
- 3. HE LLO my name is Hadley
- 4. had.co.nz/stat310
- 5. Introduction to probability
- 6. What is probability? • Mathematical machinery to deal with uncertain events • What does uncertain mean? • What is an event?
- 7. Random experiment An observation that is uncertain: we don’t know ahead of time what the answer will be (pretty common!) Ideally we want the experiment to be repeatable under exactly the same initial conditions (pretty rare!)
- 8. Sample space A set containing all possible outcomes from an experiment. Often called S. An event is a subset of the sample space
- 9. Random experiments • The sequence of dice • The length of time until rolls until you get a six your next sneeze • The weather tomorrow • My age • The next hand in a • The result of a coin ﬂip poker game • The weight of a bag of • Your ﬁnal grade in this m&m’s class • The sex of a randomly • The next President of selected member of the United States class
- 10. Your turn • How could you classify these different experiments based on the sample space? • Think (2 min) • Pair (3 min) • Square (3 min) • Share (2 min)
- 11. Contents • Numeric (quantitative) • Non-numeric (qualitative) • Will need to put both on a common framework (next week)
- 12. Cardinality • Small (< 10) • Large, but ﬁnite • Countably inﬁnite • Uncountably inﬁnite • We will follow this order as we develop increasingly complex mathematical tools
- 13. Events • An event is a subset of the sample space • Set of all possible events is the power set of S • Examples
- 14. Set algebra • Intersection and union are: • Commutative (order from left to right doesn’t matter) • Associative (order of operation doesn’t matter) • Distributive (can expand brackets) • You should be familiar with everything on: http://en.wikipedia.org/wiki/Algebra_of_sets
- 15. Terminology • Mutually exclusive • Exhaustive • Mutually exclude + exhaustive = partition
- 16. How do we deﬁne uncertainty? • Associate a probability with each element of the sample space. • Deﬁned by the function probability mass function (pmf). • The probability is the long run relative frequency
- 17. Properties of pmf • What are some properties that the pmf must have? (Use your common sense) • For example, take the random experiment of ﬂipping two coins and observing whether they come up heads or tails. How are the probabilities of the different events related?
- 18. Properties of pmf • Basic (as deﬁned by book) • Important derived properties (T 1.2-1 - T1.2-6) • Strategies of T1.2-3 and T1.2-5 particularly important