STATISTICS PREKNOWLEDGE Slides 1 - probability theory

P( A | H ) = P ( A ∩ H ) P({4,6}) 2 / 6 = = = 2/3 P( H ) 1/ 2 1/ 2 Apparently, the pre-information that H has occurred makes the probability of ‘even’ larger.
A
B
random experiment: throwing a die
Sample space {1, 2, 3, 4, 5, 6}; D = ‘low’ = {1, 2}, C = ‘high’ = {5, 6}; C ∩ D = ‘low and high’ = , C and D are disjoint.
Random variables
Setting: random experiment; outcomes in sample space ; model P. A random variable (shortly rv) is a prescript that attaches a value to each outcome of the sample space. (So, a mathematical function.) - working definition: quantity for which the actual outcome is determined by chance. - the actual outcome of a random variable is called the realisation. There are two types of rv: quantitative (the values are ordinary numbers) and qualitative (the values are categories)
Introduction probability theory
In Probability, the consequences of the presence of chance are considered. So, firstly there has to be chance. A random experiment is an experiment for which the actual outcome is determined by chance. (possible) outcomes : most basic possible results of that experiment sample space : set of all N possible outcomes Examples: - flipping a coin - throwing a die and observing the # eyes - arbitrarily choosing an element of a population - the result of a penalty (goal/miss) - next year’s profit of a company
Week 1, overview: Introduction probability theory (see Ch 6 and 7 Nieuwenhuis for more information). Discrete random variables (Ch 8): Density function, cumulative distribution function. Expectation, variance and standard deviation. Families of discrete distributions (Ch 9): Bernoulli distribution Binomial distribution
approaches a constant as n → ∞
This constant is P(A)
11
12
2
A and B are independent if: or equivalently: Example Throwing a die.
P ( A | B) = P ( A)P( A ∩ B)来自= P( A) × P( B)
Course: Statistiek voor HBO
Courseweek 1 Lecturer: Marieke Quant e-mail: M.Quant@uvt.nl Office: K 626
Course material:
Statistiek voor HBO: 14 weeks lecture (read the theory before the lecture), instruction lecture and tutorial. From week 5 onwards computer practicals. Practice yourself with weekly exercises in tutorials and practicals. In Blackboard you can find: handout lecture slides, weekly exercises tutorials and instructions classes, answers to odd exercises, datasets of the book, manual with additional exercises, computer manual, old exams. Grade: midexam (20%), TA (20%, only if this TA has not yet been made), final exam (60%, grade at least 5.0 to pass the course). A graphical/programmable calculator is not allowed; only a simple (but good) pocket-calculator is allowed (and necessary).
7
8
Setting: random experiment; sample space ; N outcomes. Central question: what do we mean by P(A), the probability that A will occur? A probability measure or model P assigns real numbers P(A) to all events A, in such a way that: P(A) ≥ 0 P() = 1 P(A∪B) = P(A) + P(B) if A, B disjoint.
Ac: complement of A (A doesn’t occur) Example random experiment: throwing a die; sample space: {1, 2, 3, 4, 5, 6} A = ‘even’ = {2, 4, 6}; B = ‘odd’ = {1, 3, 5}, B = Ac single events: {1}, {2}, , {6} C = ‘high’ = {5, 6};
a. Classical definition of probability Requirement: all outcomes are equally likely.
N ( A) (where N(A) = # outcomes in A) N Remark: this random experiment is like throwing a fair die with N sides. P ( A) =
4
Example
random experiment: flipping two coins
More definitions events A, B, …: subsets of the sample space single events; multiple events; empty event
“event A occurs”: the actual outcome falls in A
Sample space : {HH, TT, HT, TH}; a possible outcome: HH Venn diagram: useful method to represent sample space.
Example
random experiment: throwing a die
Sample space {1, 2, 3, 4, 5, 6}; A = ‘even’ = {2, 4, 6}; C = ‘high’ = {5, 6}; A ∪ C = ‘even and/or high’ = {2, 4, 5, 6}, A ∩ C = ‘even and high’ = {6}.
Example random experiment: flipping two coins. Sample space : {HH, TT, HT, TH}. Because all outcomes are equally likely: P(HH) = A = ‘at least one head’ = {HH, HT, TH}, P(A) = N(A)/ N = . Or P(A) = P(HH) + P(HT) + P(TH) = + + = . Or P(A) = 1 P(TT) = 1 = . 10
Consequence: probability that A occurs: add all probabilities of the outcomes that lead to event A.
9
b. Empirical definition of probability Requirement: the random experiment is independently and identically repeatable. (Is not always the case ....) Then the following definition of P can be used: In n trials, n(A) is the number of trials where A occurs The law of large numbers states that: The ratio