L6_rev1

# L6_rev1 - PROBABILITY Chapter 2.1 Deterministic dx = bx dt...

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PROBABILITY Chapter 2.1 Deterministic: dx dt = bx. Statistical: observation = true value + error error diﬀerent each time experiment is performed. 1

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Example. Quality control. Sample n , X = proportion defective, p = true de- fective rate. X = p + error In practice, p unknown, X observed. What information about p in X ? What can we conclude? 2
Suppose repeat sampling experiment many times. Keep track X values. Tab- ulate. Histogram. Frequencies show variability = randomness in a single X. Would be nice if cluster about true p with small variability. Why? Paddle Experiment. 3

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Example. Lewis Fry Richardson (1881- 1950) Database of 315 conﬂicts from 1820 to 1950. Poisson model (to be discussed) ﬁt well. Suggested onset of war is random process. (American Sci- entist January-February 2001). 4
Axiomatic probability: Model for de- scribing random phenomena. Many random phenomena show regu- larity in the long run, although indi- vidual outcomes cannot be predicted. This is what probability tries to de- scribe. In the Quality Control example, each sample describes a possible out- come. The frequency histogram repre- sents the empirical proportion of times each outcome occurred, or the empiri- cal probability of each outcome. Prob- ability therefore begins by considering all possible outcomes. 5

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S sample space (set of all outcomes) A,B,C,. .. subsets (called events) S A B C P ( A ) probability of A. Number reﬂect- ing our measure of how likely A is to occur. 6
Objective probability – identiﬁed with proportion in many repetitions as in Quality Control Example Subjective probability – expresses degree of belief in outcome. P ( A ) = amount willing to bet so receive 1 if A oc- curs, or 0 if

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L6_rev1 - PROBABILITY Chapter 2.1 Deterministic dx = bx dt...

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