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# lecture1 - Lecture-1 Probability theory deals with the...

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1 Lecture-1 Probability theory deals with the systematic study of random (non deterministic) phenomena, which under repeated experiments yields different outcomes. Nevertheless these experiments have certain underlying patterns about them. Example 1.1: Toss a coin (experiment) outcome is not predictable (can be head (H) or tail (T)).

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2 Suppose the same experiment is performed 1000 times. For a fair coin, we expect approximately 500 heads and about 500 tails. The notion of an experiment assumes a set of repeatable conditions that allow any number of identical repetitions. When an experiment is performed under these conditions, certain elementary events occur in different but totally uncertain ways. We can assign a nonnegative number the probability of the event in various ways: i ξ ), ( i P i
3 Laplace’s Classical Definition: The Probability of an event A is defined a-priori without actual experimentation as provided all these outcomes are equally likely. Consider a box with n white and m red balls. In this case, two elementary outcomes: white ball or red ball. Probability of selecting a white ball at random , outcomes possible of number Total to favorable outcomes of Number ) ( A A P = . m n n + =

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4 Relative Frequency Definition : The probability of an event A is defined as where n A is the number of occurrences of A and n is the total number of trials. The axiomatic approach to probability, due to Kolmogorov, developed through a set of axioms (below) is generally recognized as superior, as it provides a solid foundation for complicated applications. n n A P A n lim ) ( =
5 The totality of all known a priori , constitutes a set , the set of all experimental outcomes. has subsets Recall that if A is a subset of , then implies From A and B, we can generate other related subsets etc. , i ξ {} ± ± , , , , 2 1 k = (1) A . . , , , ± C B A , , , , B A B A B A {} { } {} A A B A B A B A B A = = = | and | or | (2) (3) (4)

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6 A B B A A B A B A A • If the empty set, then A and B are said to be mutually exclusive. • A partition of is a collection of mutually exclusive subsets of such that their union is . , φ = B A . and , 1 = = = ± i i j i A A A B A = B A 1 A 2 A n A i A A (5) j A
7 De-Morgan’s Laws: B A B A B A B A = = ; A B B A A B B A A B B A A B It is meaningful to talk about at least some of the subsets of as events, for which we must have a mechanism to compute their probabilities. Example 1.2: Consider the experiment where two coins are simultaneously tossed. The various elementary events are (6)

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8 {} . , , , 4 3 2 1 ξ = ) , ( ), , ( ), , ( ), , ( 4 3 2 1 T T H T T H H H = = = = and The subset is the same as “Head has occurred atleast once” and qualifies as an event. Suppose two subsets
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## This note was uploaded on 04/24/2011 for the course ECE 6303 taught by Professor Voltz during the Spring '11 term at NYU Poly.

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lecture1 - Lecture-1 Probability theory deals with the...

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