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Unformatted text preview: 26 1.6 One Useful Discrete Distribution: The Binomial Distribution
Text: Read pages 4851 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 Nonexamples: 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, 6sided 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)
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32 41.?
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 Spring '08
 DeVasher
 Statistics, Probability theory, Discrete probability distribution, Jill, Geckos, binomial variable

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