131A_1_Lecture_3-2_Winter_2012

131A_1_Lecture_3-2_Winter_2012 - UCLA EE131A (KY) 1 EE 131A...

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Unformatted text preview: UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 3 - 2 UCLA EE131A (KY) 2 Lecture 3 -2: Axioms of probability (1) The theory of modern probability was developed formally by A.N. Kolmogorov of the Univ. of Moscow in the 1920s from a set of axioms (such as Euclidean geometry and other branches of mathematics). From the three axioms of probability theory, and by using logical deductions and standard mathematical operations, we can obtain all known results in probability. UCLA EE131A (KY) 3 Axioms of probability (2) In order to obtain the maximum intuition in understanding basic probability concepts, we can use Venn diagram (as shown below) as a simple graphical way to interpret the probability of events. Specifically, we can intuitively equate the probability of an event A as the area of A in the Venn diagram, i.e., P( A ) = Area( ). B S A UCLA EE131A (KY) 4 Axioms of probability (3) Consider a random experiment with a sample space S . A probability function P( E ) defined for all events E in S must satisfy the following three axioms: I. 0 P( A ) ~ Probability function is a non- negative valued function. From the Venn diagram point of view, 0 P( A ) is consistent with 0 Area(A), since the area of any object is non-negative valued. UCLA EE131A (KY) 5 II. P( S ) = 1 ~ Probability function takes values over [0, 1]. No probability function...
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131A_1_Lecture_3-2_Winter_2012 - UCLA EE131A (KY) 1 EE 131A...

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