1. 1. Probability 2. Random variables 3. Inequalities 4. Convergence of random variables 5. Point and interval estimation 6. Hypotheses testing 7. Nonparametric inference Statistics 1

2. 1. Probability  Introduction  Uniform sample spaces. Laplace’s rule  Independent events  Conditional probability  Total probability theorem  Bayes’ theorem 2

3. PROBABILIDAD Introduction Random experiments verify:  Each result is not known beforehand.  All possible results are known in advance.  Can be repeated in the same conditions. 3 Elements:  Sample space  Events  Probability

4. PROBABILITY Introduction  Sample space (  ): Set of possible outcomes of an experiment.  Events: Subsets of the sample space. Union: A  B Intersection: A  B Complement of A: A C 4

5. Introduction  Probability: P assigning a real number P(A) to each event A, verifying: (i)0  P(A)  1 (ii)P(  )=1 (iii){A i } disjoint  P(  A i )=  i P(A i ) A i, A j disjoint if A i  A j =  Properties (i)P(A C ) = 1 - P(A) (ii)P(  ) = 0 (iii)A  B  P(A)  P(B) (iv)P(A  B) = P(A) + P(B) - P(A  B) 5 PROBABILITY

6. Uniform sample spaces  ={ω 1,...,ω n } is a uniform sample space if all the elements have the same probability.  = {ω 1 }  {ω 2 } ...  {ω n } Since P(  ) =1, P(ω i ) =1/n. 6 PROBABILITY

7. Laplace’s rule A={ω 1,...,ω r } ; P(A) = P( {ω 1 }  {ω 2 } ...  {ω r } ) = =1/n +... + 1/n = r/n P(A)= favorable outcomes / possible outcomes Remark: only for uniform sample spaces. 7 r PROBABILITY

8. Independent events A and B are independent if P(A  B) = P(A)  P(B) Remark: P(A  B) = P(AB) 8 PROBABILITY

9. Conditional probability A and B; P(B) > 0; P(A | B) = P(A  B) / P(B) Multiplication law: P(A  B) = P(A | B)  P(B) Property If A and B are independent then P(A | B) = P(A) 9 PROBABILITY

10. Total probability theorem Let with B is any event. Then: 10 B A4A4 A3A3 A2A2 A1A1 AiAi A7A7 A6A6 A5A5 PROBABILITY

11. Bayes’ theorem Let with B any event. Then : 11 PROBABILITY