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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!