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Notes_5_with_comments - 26 1.6 One Useful Discrete...

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Unformatted text preview: 26 1.6 One Useful Discrete Distribution: The Binomial Distribution Text: Read pages 48-51 Two types of variables: Discrete and Continuous o A variable is discrete if its set of possible values is either finite or can be listed as an infinite sequence (e.g., l, 2, 3, 4, 5, ...). o A discrete variable x is specified by a mass function p(x) satisfying: (1) p(x) Z 0 for all x, and (2) Zp<x)=1. 0 There are special variables that occur frequently in experimentation and otherwise (such as the normal variable); one important discrete variable is the binomial variable Definition: A binomial variable is the number of successes x in n independent and identical trials of an experiment with only two possible outcomes. Each trial has probability of success It. The important assumptions for a distribution of a variable x to be binomial: (1) The variable x is a count of the number of successes out of the finite number of trials n. (2) The n trials are identical and independent. (3) There are 2 possible outcomes for each trial: “success” or “failure.” (4) The proportion of time an individual trial is a success is 7r. Examples: 0 Number of free throws that you make (successes) out of 5 tries 0 Number of defective parts (“successes”) that you pick out of 1000 parts, where the probability of a defective is 0.02 0 Number of heads (“successes”) you obtain when you flip a coin 6 times 0 Number of “bumpy side” Lego (“successes”) lands out of 10 trials Non-examples: Why? Number of free throws you take until a basket is made —-‘a '40 4*; 0+ f“ j; Number ofchildren a couple has ‘6’ no 36+ ‘it (57‘ 7‘1‘ J1 Number of problems you get correct on your first homework set (out of 10)~—-» «tie propor‘l‘wv 6% / _ Sucua}§(’3 1) 910+ c.043i‘znl’ r The variable x that equals the number of successes in n independent and identical trials of an experiment, where the probability of success of each trial is 7r, is a binomial variable. We say that n, the number of trials, and 7r, 0 s n S 1, the probability of success, are parameters of the binomial distribution. The mass function of a binomial variable x is: n p(x) = proportion ofx successes out ofn trials = [ )n‘fl — 7r)"‘x , x = O, 1, 2, ..., n. x n Definition: [ J x Note: For a binomial random variable x, p(x) Z O for all x, and Zp(x) = 1. We can check in Maple: >assume(p > O); assume (p < 1); >f := binomial(n, x) * p " x * (l — p) " (n — x); (11»,1’) fz= binomial (n, x) p3 (l ~p~) >simplify(sum(f, x = .. n)); n where “binomial(n, x)” is Maple’s notation for [ j x Example 1. Each time you buy a candy bar from the vending machine by the campus mail center, you have a 0.9 probability of receiving the item you choose. Assume each vending machine trial is independent. Let the discrete variable x represent the number of correct items you receive when choosing 3 different items from the machine. If x is the number of correct items you receive, what is the probability 7r of a success on a given trial? [trzv‘iO If y (another variable) is the number of incorrect items you receive, what is the probability of a “success?” ‘ a l D How many trials n are there? Are the trials independent and identical? A j 3 i’1A37‘7’tLfli “Ci (Kimch Is x a binomial variable? Why or Why not? Is y a binomial variable? ‘h l) EH 0% '2‘”ri 3) Cmisfi/IiF/OPU/‘km of “Success?” 3> “[1507.Q>5,L//7€ oyx‘i’cmej «i» @cht ifij l 28 Example 2. Snoopy is an 80% free throw shooter. If he takes 4 shots and x is the number he makes out of 4, determine the mass function p(x). (a) If Snoopy always warms up by shooting 4 shots, what proportion of the time will he make exactly two shots? MM: (WWW (0 WW 1 (b) What proportion of the time will he make at least one shot? j> .2: i) : PWX =0) Is my; 3)0_/,2)7 : I~ /«/e,ooié): Example 3. For the leopard gecko (Eublepharis macularius), the gender of their offspring is determined by the temperature during embryonic development. At 30° C, the offspring is almost completely female. At 32.5° C the offspring is almost completely male. Researchers at the University of Texas have determined that at 31° C, the proportion of males produced is 35%. They have 20 leopard gecko embryos that have been incubated at 31° C and will soon hatch. Suppose they are interested in the number of male leopard geckos from this litter of 20. What proportion of litters of size 20 would contain at least 5 male geckos? [Hint Maple or calculator.] Wm): (*:?>g.5yy(;tgsyv * ("25> my mu + x O f (BMW) [1») r : K S WM” “ "‘[mfl/“iv/“Wf— ‘* {iii/,itmti 1: l" .AHYZ thifism 29 Suggested Practice Problems/Board Problems 1. Is it binomial? Determine for which, if any, of the following scenarios, the distribution of x can be adequately modeled by the binomial distribution. If it is binomial, determine n and 71?. (3) Suppose we toss a fair, 6-sided die 10 times. Let x be the number of “l ’s” we get in 10 tosses. Binomial? Ifyes, determine n and TC: I N10 M, j A 1: I T: } (b) Jack and Jill are not so good at basketball (they misspent their youth fetching water), but Jill plays anyway even though she only makes 20% of her shots. Let x be the number of shots Jill takes before she makes a basket. Binomial? If yes, determine n and 1t: 5 N or D i Molt, '4 L (c) Many component manufacturers ship out their product in batches. They can use a quality control method called acceptance sampling in which a small sample of components in each batch of components is carefully checked. If the number of defectives in the sample does not exceed a certain number, then the batch is shipped. Let x be the number of defectives in a sample of size 10 from a batch of 400 components where 1% are defective. Binomial? Ifyes,determinenandn: / (I: 3 "Pt-“'3 2. Suppose you have a true/false current events quiz in your humanities course. It is based upon the nightly news from the previous evening. You did not have time to watch the nightly news; therefore, you had to guess the answers to all 5 questions. You need to answer at least 3 out of the 5 questions correctly to pass the quiz. In the long run, what proportion of these types of quizzes (with 5 questions) & your guessing method will yield at least 3 correct answers? Is it likely that you’ll pass the quiz? )(2 a a‘i comet 3463585 3mm) X MBIN0H14L(": 3’) Tzé) are, (WWW . ('22) (W In the long run, what proportion of these types of quizzes (with 5 questions) & your guessing method will yield at least one correct answer from you? [Hint You don’t need a calculator for thisl] pay) I~ Ward) Iv P0010) ll .' ,L 32 to +3’ +1] ' ..a r. ,-(;)(t)°(l,)‘g ;~(/Yl)(iri: 33”: : F0633 ‘l’ TP/qul +‘P(X:§) New are“ EX?“ [:9 32 41.? 2 nor Tm umy/ ...
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Notes_5_with_comments - 26 1.6 One Useful Discrete...

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