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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 1920’s 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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