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Chapter 4 Notes - Chapter 4 Probability Distributions 1...

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Chapter 4: Probability Distributions 1 Random Variables } Random Variable X associates a numerical value to each outcome of an experiment. (r.v.) Example 1: X: # heads obtained in three tosses of a fair coin. Outcome X HHH 3 HHT 2 HTH 2 HTT 1 THH 2 THT 1 TTH 1 TTT 0 Value of X Event X = 0 { TTT } X = 1 { HTT, THT, TTH } X = 2 { HHT, HTH, THH } X = 3 { HHH } P ( X = 0) = 1 8 P ( X = 1) = 3 8 P ( X = 2) = 3 8 P ( X = 3) = 1 8 } A r.v. is discrete if its set of possible values is a discrete set. } A r.v. is continuous if it represents some measurement on a continuous scale. eg) X: # heads obtained in three tosses of a coin: continuous. X: amount of precipitation produced by a storm: discrete. 2 Probability distribution of a discrete r.v. } Probability distribution of a discrete r.v.: f ( χ ) = P ( X = χ ) } f ( χ ) 0 for all χ allχ f ( χ ) = 1 Example 2: 1
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(a) f ( χ ) = χ - 2 2 for χ = 1 , 2 , 3 , 4 f (1) = - 1 2 not a probability distribution. (b) f ( χ ) = χ 2 25 for χ = 0 , 1 , 2 , 3 , 4 f ( χ ) 0 for all χ , but 4 χ =0 f ( χ ) = 0+ 1 25 + 4 25 + 9 25 + 16 25 = 30 25 = 6 5 6 = 1 not a probability distribution. Example 3: 30% of the trees in a forest are infested with a parasite. Four trees are selected at random. X: # trees sampled that have the parasite. Let I: infested, N: not infested. X=0 X=1 X=2 X=3 X=4 NNNN NNNI NNII NIII IIII NNIN NINI INII NINN NIIN IINI INNN INNI IIIN ININ IINN P ( X = 0) = P (NNNN) = (0 . 7) 4 = 0 . 2401 P ( X = 1) = 4 · (0 . 7) 3 · (0 . 3) = 4 · (0 . 1029) = 0 . 4116 P ( X = 2) = 6 · (0 . 7) 2 · (0 . 3) 2 = 0 . 2646 P ( X = 3) = 4 · (0 . 7) · (0 . 3) 3 = 0 . 0756 P ( X = 4) = (0 . 3) 4 = 0 . 0081 0 1 2 3 4 0.2 0.4 f(X) } Probability model: An assumed form of the probability distribution that describes the chance behavior for a r.v. X.
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