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Chapter 3

# Chapter 3 - IENG 213 Probability and Statistics for...

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Click to edit Master subtitle style IENG 213: Probability and Statistics for Engineers Instructor: Steven E. Guffey, PhD, CIH © 2002-2009 Chapter 3: Probability Distributions

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22 Background Random variable: function that associates a real number with each element in the sample space Example: x = number of tails when flip coin 3 times Example: x = height of each person in this class
33 Sample Space Discrete: A sample space containing a finite number of possibilities or an unending sequence with as many elements as there are numbers. Variable called a “discrete random variable” Can be counted Example: No. people in the room with red shoes on. Continuous sample space: Sample space containing an infinite number of possibilities equal to the number of points on a line segment. Non-discrete Variable is a “continuous random variable” Can’t be counted but can be measured Example: heights of children

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44 3.2 Discrete Probability Distributions Discrete random variable has a certain probability of equaling each of its possible values Example: tossing coin 3 times x = number of heads S = {HHH,HHT, HTH, THH, TTH, THT, HTT, TTT} For x=2, p= Using formulae f(x)= P(X=x) f(3) =
55 3.2 Probability Distribution for Discrete Variables Set of ordered pairs (x, f[x] ) is a probability function, probability mass function, or probability distribution of the discrete random variable, X, if: f(x) > 0 Σ f(x) = 1 P(X=x) = f(x)

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66 Example 3.3 A shipment of 8 computers contains 3 that are defective. If we purchase 2, find the probability distribution for the number of defectives (N x ) we might receive. 28 ) ! 6 ( 2 ) ! 6 )( 7 ( 8 )! 2 8 ( ! 2 ! 8 )! ( ! ! ! ! = = - = - = = r n r n r n N total ( 29 ( 29 ! ! !( )! ! ! good bad X x bad good n n N x n x r x n r x = = - - - - Need to find N x and Ntotal so can use P(x) = N x / N total
77 Example 3.3-continued A shipment of 8 computers contains 3 that are defective. If we purchase 2, find the probability distribution for the number of defectives. Pr(x) = N x /N total 28 = total N ( 29 ( 29 15 5 3 )! 1 5 ( ! 1 ! 5 )! 1 3 ( ! 1 ! 3 1 5 ! 1 ! 3 1 = = - - = = = X N ( 29 ( 29 3 1 3 )! 0 5 ( ! 0 ! 5 )! 2 3 ( ! 2 ! 3 0 5 2 3 2 = = - - = = = X N 0 10 Pr 28 X n N = = = x 0 1 2 f(x) A shipment of 8 computers contains 3 that are defective. If we purchase 2, find the probability distribution for the number of defectives we might receive. Pr(x) = N x / N total ( 29 ( 29 10 10 1 )! 2 5 ( ! 2 ! 5 )! 0 3 ( ! 0 ! 3 2 5 ! 0 ! 3 0 = = - - = = = X N 1 15 Pr 28 X = = 2 3 Pr 28 X = =

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88 Cumulative Distributions for Discrete < < - = x for t f x F x t ), ( ) ( Example: for random variable M, F(2) = P(M ) = f( ) + f( ) + f( ) Remember: F(x) defined for all real numbers, not just for the values of x we happen to choose.
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• Spring '08
• STAFF
• Probability distribution, Probability theory, probability density function, Cumulative distribution function

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Chapter 3 - IENG 213 Probability and Statistics for...

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