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Classnotes_4 - 2 Random Experiments • D fi i i Definition...

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Random Experiments D fi i i • Definition: 3 Sample Spaces D fi i i • Definition: E.g. if we are to flip a (typical) coin once, the possible outcomes are “heads” or “tails” ( that we possible outcomes are heads or tails ( that we can denote H and T respectively ) to give: S = { H S = { H , T } If our experiment involves flipping a single coin t i t twice we get: S = { HH, HT, TH, TT } 4
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More “Sample Space” Examples Example 2-1 5 More “Sample Space” Examples Example 2-1 (continued) 6
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Discrete v Continuous Sample Discrete v. Continuous Sample Spaces Discrete : Th b f b f th l i fi it The number of members of the sample space is finite or C ti Continuous : The number of members of the sample space is infinite d and uncountable Notes: A “ countably infinite ” set has the same number of members as the set of positive integers (i.e. you can count them but it’ll take you forever ) An uncountable set has the same number of members as a portion of the An set has the same number of members as a portion of the real line (e.g. there’s an uncountable number of real numbers between 0 and 1 ) 7 More “Sample Space” Examples Example 2-2 8
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More “Sample Space” Examples Example 2-2 (extended) Where “n” means a failed connector and “y” an acceptable one In this case, the sample space is discrete (as the number of outcomes is countably infinite ) 9 The Outcomes in a Sample Space The Outcomes in a Sample Space Are Not Necessarily Equally Likely In general, the outcomes in a sample space are not equally likely: If the coin used in our coin-flipping examples is “ fair ”, each outcome is equally likely In all of our other examples so far, the outcomes are not equally likely Another example with unequally likely outcomes: Suppose we have a population of 100 items, n of which are defective, and we draw a sample of two items D i d” i b “ ” d d f i i b “d” h l Denoting a “good” item by “g” and a defective item by “d”, the sample space is as follows ( provided that n>=2 ): S = {gg, gd, dg, dd} ... if the ordering of the samples is important or S = {gg, gd, dd} ... if the ordering of the samples is unimportant Q. what’s the sample space if there is only a single defective item in the population? 10
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Sampling With (and Without) Sampling With (and Without) Replacement In experiments where samples are taken from a population, it is significant whether or not this is done with or without it is significant whether or not this is done with or without replacement: – When sampling with replacement , the sampled item is replaced When sampling
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