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Unformatted text preview: STAT 100A Review for midterm Note: The following are the materials to be covered in the midterm. 1 Basic concepts When an experiment is performed, the outcome can be random. The sample space is the set of all the possible outcomes. It is often denoted by Ω. An event is a subset of the sample space, and it is often denoted by A , B , C , etc. An event is often expressed as a statement about the outcome, and it is the set of the outcomes that satisfy this statement. Probabilities are defined on the events. 2 Equally likely examples We have strong intuitions or common sense about probabilities when the outcomes are equally likely. (1) Flip a fair coin n times independently, then all 2 n sequences are equally likely. (2) Roll a fair die once, then all the 6 numbers are equally likely. Roll a fair die twice indepen- dently, then all the 36 pairs of numbers are equally likely. (3) Generate a random number uniformly from [0, 1], then all the numbers in [0, 1] are equally likely. Generate two uniform random numbers independently, then all the points in the unit square [0 , 1] 2 are equally likely. For equally likely cases, P ( A ) = | A | / | Ω | , where | A | is the number of elements in A in the discrete case, or the length, area, or volume of A in the continuous case. 3 Relations There are three relations of events. (1) Intersection-AND. Notation: A ∩ B . It includes elements that are in both A and B . Two events A and B are disjoint or mutually exclusive if A ∩ B = φ , where φ is the empty set. See Figure 1 for illustration. Figure 1: Intersection. (2) Union-OR. Notation: A ∪ B . It includes elements that are in either A or B . See Figure 2 for illustration. (3) Complement-NOT. Notation: ¯ A . It includes elements that are not in A . See Figure 3 for illustration. 1 Figure 2: Union. Figure 3: Complement. 4 Axioms The following are the three axioms of probability. (1) For any A ⊂ Ω, P ( A ) ≥ 0. (2) P (Ω) = 1. (3) Additivity: If A ∩ B = φ , then P ( A ∪ B ) = P ( A ) + P ( B ). The axioms cannot be proved. They are assumed, based on our common sense of probability. 5 Interpretations The following are three intuitive interpretations of probability. (1) Geometric interpretation. Suppose we randomly throw a point into a region Ω, then the probability that the point falls into a sub-region A is P ( A ) = | A | / | Ω | , where | A | is the area of A . See Figure 4 for illustration. Figure 4: Throw a random point into Ω. (2) Population interpretation. Suppose we have a population of M people. r of them are red, and b of them are blue, and r + b = M . If we randomly sample a person, then the person is red is r/M ....
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This note was uploaded on 01/29/2010 for the course STATS 100A 262303210 taught by Professor Wu during the Fall '09 term at UCLA.
- Fall '09