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Unformatted text preview: 1 Elements of Probability and Statistics 2 Probability and Random Variables • Random Experiment – An experiment whose outcomes are not known in advance but the set of all possible outcomes is known. • Sample Space S – Set of all possible outcomes. • Sample Point (Elementary Event) – Each element of sample space. 3 Example – If the random experiment consists of tossing a fair coin again and again until the first head appears, the sample space can be represented by A sample space is discrete if the number of sample points are finite or countable, and continuous if sample points consists of all the numbers on some finite or infinite interval of the real line. An event is a subset of the sample space. { , , ,......} S H TH TTH = 4 Probability Measure – Probability P is a set function defined on the class of all events to R, the set of real numbers, such that Theorem – Let P be a probability measure defined on a sample space S, then ∞ ∞ ≤ ≤ ∑ U 1 2 n n n=1 n=1 (i) 0 P(A) 1,foreveryevent A. (ii) P(S)=1. (iii) If theeventsA ,A ,...aremutuallyexclusive,then P( A )= P(A ). φ ∪ ∩ ⊆ ⇒ ≤ c (a) P( ) = 0 (b) P(A) = 1 P(A ) (c) P(A B) = P(A) + P(B)  P(A B) (d) A B P(A) P(B). 5 Conditional Probability – To calculate the probability that A occurs, given that B has occurred, means reevaluating the probability of A with the information that B has occurred. Thus, B becomes new sample space and we are interested in the part of A that occurs with B, that is, . Thus, we must have the formula provided that ∩ A B ( ) (  ) ( ) A B A B B ∩ = P P P ( ) B ≠ P 6 Multiplication Rule – For events A and B, Law of Total Probability – ( ) ( ) (  ), ( ) 0, ( ) ( ) (  ), ( ) 0. A B A B A if A and A B B A B if B ∩ = ≠ ∩ = ≠ P P P P P P P P 1 2 1 2 1 1 2 2 , ,..., ( ) , ( ) ( ) 0, 1,2,..., ( ) ... . , ( ) ( ) (  ) ( ) (  ) ... ( ) (  ). n i j i n n n A A A a A A if i j b A i n c A A A S A A A A A A A A A A A φ ∩ = ≠ = ∪ ∪ ∪ = = + + + Let beeventssuchthat(partitionof S) P Then,foranyevent P P P P P P P 7 Problem: A factory uses 3 machines X, Y and Z to produce certain items. Suppose machine X produces 50% of the items of which 3% are defective, machine Y produces 30% of the items of which 4% are defective and machine Z produces 20% of the items of which 5% are defective. Find the probability that a randomly selected item is defective . 8 Independent Events – Two events A and B are said to be independent if ( ) ( ) ( ). A B A B ∩ = P P P Events are said to be mutually independent if Bayes’ Theorem – 1 2 , ,..., n A A A 1 2 1 2 ( ... ) ( ) ( )... ( ) for 1,2,..., j j A A A A A A j n ∩ ∩ ∩ = = P P P P 1 ( ) (  ) (  ) , 1,2,..., ....
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 Spring '11
 ritadubey
 Statistics, Probability, Probability theory, Probability mass function

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