132B_1_Recitation1_Probability_Review

132B_1_Recitation1_Probability_Review - EE132B - Recitation...

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EE132B - Recitation 1 Probability review Prof. Izhak Rubin rubin@ee.ucla.edu Electrical Engineering Department UCLA 1
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Probability space • A probability space is defined by three parameters (V, E, P). – V: represents the sample space . • Sample space : the set of all the outcomes w. – E: represents the collection of subsets of V called events . An event is a set of outcomes. The set of events is E. – P: is a function of V that maps events to the interval [0, 1], i.e., P assigns probabilities to different events in E. 2
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Random variable • Definition: A random variable X is a function that associates a real number with each elements of the sample space, i.e., (in more technical terms,) maps V R , or, assigns a value X(w) = R to each outcome w. 3
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Example: Toss a coin V = {H, T} E = all possible combinations of V = {font, (H), (T), (H,T)} P: P(font) = 0 P(H) = ½ P(T) = ½ P(H, T) = 1 • An example of a random variable can be: X(H) = +1 (if you get a head, you win one dollar) X(T) = -1 (if you get a tail, you lose one dollar) 4
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Three axioms of probability • Any probability function must obey the following: – 1. P(A) >= 0 – 2. P(AUB) = P(A) + P(B) iff A and B are disjoint – 3. P(V)=1 5
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132B_1_Recitation1_Probability_Review - EE132B - Recitation...

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