Sec$on 6.1 - wghsvoinea.weebly.com€¦ · We see that 1 in every 1000 babies would have an Apgar...
Transcript of Sec$on 6.1 - wghsvoinea.weebly.com€¦ · We see that 1 in every 1000 babies would have an Apgar...
Sec$on6.1:
DiscreteandCon.nuousRandomVariables
Tuesday,November14th,2017
Learning Objectives After this section, you should be able to:
The Practice of Statistics, 5th Edition 2
ü COMPUTE probabilities using the probability distribution of a discrete random variable.
ü CALCULATE and INTERPRET the mean and standard deviation of a discrete random variable.
Thursday:
ü COMPUTE probabilities using the probability distribution of certain continuous random variables.
Discrete and Continuous Random Variables
Introduc$on:Supposewetossafaircoin3$mes.
1. Whatisthesamplespaceforthischanceprocess?Howmanyoutcomesarepossible?Whatisthelikelihoodofeachoutcome?
SampleSpace:HHHHHTHTHHTTTHHTHTTTHTTT
Thereare8equallylikelyoutcomesinoursamplespace.
Theprobabilityis1/8foreachoutcome.
2.LetX=numberofheadsobtainedinatoss.ThevalueofXwillvaryfrom____,____,_____,or_____.HowlikelyisXtotakeeachofthosevalues?
10 32
3.FindandinterpretP(X≥1).
Thisprobabilitymeansthatifweweretorepeatedlyflip3coinsandrecordthenumberofheads,wewillget1ormoreheadsabout87.5%ofthe$me.
Whatisarandomvariable?Whatarethetwomaintypesofrandomvariables?
Anumericalvaluethatdescribestheoutcomes
ofachanceprocess.
DISCRETERANDOMVARIABLES
CONTINUOUSRANDOMVARIABLES
Whatisaprobabilitydistribu.on?
Aprobabilitydistribu$onisamodelthatgivesthepossiblevaluesandprobabili$esofarandomvariable.
Canbegivenastablesorhistograms.
Whatisadiscreterandomvariable?Givesomeexamples.
• Arandomvariablewithgapsbetweenthevalues
• Discreterandomvariablesareo=en>mesthingsyoucancount.
• Forexample,shoesizevs.actuallengthoffoot..Whichoneisdiscrete?
EXAMPLE1:ApgarScores(BabiesHealthatBirth)
a)Showtheprobabilitymodelislegi.mate.
• Alloftheprobabili.esfortheApgarscoresarebetween0and1.
• Theprobabili.esaddupto1.• Therefore,wecanconcludethatthisisalegi.mateprobabilitymodel.
b)Makeahistogramoftheprobabilitydistribu.on.Describewhatyousee.
• Shape:thedistribu.onappears
skewedleXandsingle-pinked.ArandomlyselectednewbornismostlikelytohaveahigherApgarscore,meaningtheyarehealthy.
• Center:ThemedianApgarscoreappearstobeascoreof8.
• Spread:Apgarscoresvarybetween0and10,althoughmostofthescoresarebetween6and10.
c)DoctorsdecidedthatApgarscoresof7orhigherindicateahealthybaby.What’stheprobabilitythatarandomly
selectedbabyishealthy?
Wewouldhaveabouta91%chanceofselec$ngahealthynewborn.
READP.350-351
Mean (Expected Value) of a Discrete Random Variable
The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability.
Suppose that X is a discrete random variable whose probability distribution is
Value: x1 x2 x3 … Probability: p1 p2 p3 …
To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: ∑=
+++==µ
ii
x
pxpxpxpxXE ...)( 332211
d)ComputethemeanoftherandomvariableX.Interpretthisvalueincontext.We see that 1 in every 1000 babies would have an Apgar score of 0; 6 in every 1000 babies would have an Apgar score of 1; and so on. So the mean (expected value) of X is: The mean Apgar score of a randomly selected newborn is 8.128. This is the average Apgar score of many, many randomly chosen babies.
PG.352-353
Standard Deviation of a Discrete Random Variable The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data.
Suppose that X is a discrete random variable whose probability distribution is
Value: x1 x2 x3 … Probability: p1 p2 p3 …
and that µX is the mean of X. The variance of X is
€
Var(X) =σX2 = (x1 −µX )
2 p1 + (x2 −µX )2 p2 + (x3 −µX )
2 p3 + ...
= (xi −µX )2 pi∑
To get the standard deviation of a random variable, take the square root of the variance.
e)Computeandinterpretthestandarddevia.onoftherandomvariableX.
The standard deviation of X is σX = √(2.066) = 1.437. A randomly selected baby’s Apgar score will typically differ from the mean (8.128) by about 1.4 units.
Now You Try!!Imagine selecting a U.S. high school student at random. Define the random variable X = number of languages spoken by the randomly selected student. The table below gives the probability distribution of X, based on a sample of students from the U.S. Census at School database. !!!!
* 5 randomly selected students will write and present their answers to the class on the whiteboard *
Showthattheprobabilitydistribu.onforXislegi.mate.
Alltheprobabili.esarebetween0and1andtheyaddto1,sothisisalegi.mateprobabilitydistribu.on.
Makeahistogramoftheprobabilitydistribu.on.Describewhatyousee.
Shape:Thehistogramisstronglyskewedtotheright.Center:Themedianis1,butthemeanwillbeslightlyhigherduetotheskewness.Spread:Thenumberoflanguagesvariesfrom1to5,butnearlyallofthestudentsspeakjustoneortwolanguages.
(c)Whatistheprobabilitythatarandomlyselectedstudentspeaksatleast3
languages?P(X≥3)=P(X=3)+P(X=4)+P(X=5)=0.065+0.008+0.002=0.075.Thereisa0.075probabilityofrandomlyselec$ngastudentwhospeaksthreeormorelanguages.
(d)ComputethemeanoftherandomvariableXandinterpretthisvaluein
context.
(e)Computeandinterpretthestandarddevia.onoftherandomvariableX.
ExitTicket:ChecksforUnderstanding
CompletethefollowingCheckYourUnderstandingProblemsonbinderpapertoturnin:• Pg.350#1-3• Pg.355#1-2
Bynextclass:
READ355-357aboutcon.nuousrandomvariablesandfinishthenotes.
Chapter5Quiz:Thursday!
• HW#21isdueThursday• CompletetheChapter5Prac.ceTestonyourown
• Comebytotutorialforextrahelp!