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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 . 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 | / | | , 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 = , where 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 , P ( A ) 0. (2) P () = 1. (3) Additivity: If A B = , 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 , then the probability that the point falls into a sub-region A is P ( A ) = | A | / | | , where | A | is the area of A . See Figure 4 for illustration. Figure 4: Throw a random point into . (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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- Fall '09