Chapter 2. Random Variables and Probability Distributions

Transcription

1 Rndom Vriles nd Proility Distriutions- 6 Chpter. Rndom Vriles nd Proility Distriutions.. Introduction In the previous chpter, we introduced common topics of proility. In this chpter, we trnslte those concepts into mthemticl frmework. We invoke lger for discrete vriles nd clculus for continuous vriles. Every topic in this chpter is presented twice, once for discrete vriles nd gin for continuous vriles. The nlogy in the two cses should e pprent nd should reinforce the common underlying concepts. There is, in the second hlf of the chpter, nother dupliction of concepts in which we show tht the sme process of trnsltion from the lnguge of proility to tht of mthemtics cn e performed not only when we hve single vrile of interest, ut lso when we hve two vriles. Agin, high-lighting this nlogy etween single nd joint proility distriutions eplicitly revels the common underlying concepts... Rndom Vriles & Smple Spces We egin with the introduction of necessry voculry. Rndom vrile A rndom vrile is function tht ssocites numer, integer or rel, with ech element in smple spce. Discrete Smple Spce If smple spce contins finite numer of possiilities or n unending sequences with s mny elements s there re whole numers, it is clled discrete smple spce. Emple..: You flip two coins. Y is rndom vrile tht counts the numer of heds. The possile results nd the vlue of the rndom vrile ssocited with ech result re given in the following tle.

2 Rndom Vriles nd Proility Distriutions - 7 result y HH HT TH TT This smple spce is discrete ecuse there re finite numer of possile outcomes. Emple..: You suject o contining N devices to test. Y is rndom vrile tht counts the numer of defective devices. The vlue of the rndom vrile rnges from (no defects) to N (ll defects). This smple spce is discrete ecuse there re finite numer of possile outcomes. Continuous Smple Spce If smple spce contins n infinite numer of possiilities equl to the numer of points on line segment, it is clled continuous smple spce. Emple..: You drive cr with five gllons of gs. Y is rndom vrile tht represents the distnce trveled. The possile results re infinite ecuse even if the cr verged miles per gllon, it could go. miles,.,.,.,. miles. The smple spce is s infinite s rel numers... Discrete Proility Distriution Functions (PDFs) Proility distriution function (PDF) The function, f() is proility distriution function of the discrete rndom vrile, if for ech possile outcome, the following three criteri re stisfied. (.) P( ) f ( ) The PDF is lwys non-negtive. The PDF is normlized, mening tht the sum over ll vlues of discrete PDF is unity. The PDF evluted t outcome provides the proility of the occurrence of outcome. Emple.4.: Eight devices re shipped to retil outlet, of which re defective. If consumer purchses computers, find the proility distriution for the numer of defective devices ought y the consumer. In order to solve this prolem, first define the rndom vrile nd the rnge of the rndom vrile. The rndom vrile,, is equl to the numer of defective devices ought y the

3 Rndom Vriles nd Proility Distriutions- 8 consumer. The rndom vrile,, cn tke on vlues of,, nd. Those re the only numer of defective devices the consumer cn uy, given tht they re only uying two devices. The net step is to determine the size of the smple spce. The numer of wys tht cn e 8 tken from 8 without replcement is 8. We use the formul for comintions ecuse the order of purchses does not mtter. This is the totl numer of comintions of devices tht the consumer cn uy. Third, the proility of prticulr outcome is equl to the numer of wys to get tht outcome over the totl numer of wys: P( ) wys of getting totl wys f () P( ) 8 f () P( ) 8 f () P( ) In ech of these cses, we otined the numertor, the numer of wys of getting outcome, y using the comintion rule nd the generlized multipliction rule. There re wys of choosing defective devices from defective devices. There re wys of choosing (-) - good devices from good devices. We use the generlized multipliction rule to get the numer of wys of getting oth of these outcomes in the numertor. As preview, we will come to discover tht this proility distriution is clled the hypergeometric distriution in Chpter 4. So we hve the PDF, f(), defined for ll possile vlues of. We hve solved the prolem. Note: If someone sked for the proility for getting (or ny numer other thn,, or ) defective devices, then the proility is zero nd f(). Testing discrete PDF for legitimcy If you re sked to determine if given PDF is legitimte, you re required to verify the three criteri in eqution (.). Generlly, the third criterion is given in the prolem sttement, so you only hve to check the first two criteri.

4 Rndom Vriles nd Proility Distriutions - 9 The first criteri,, cn most esily e verified y plotting f() nd showing tht it is never negtive. The second criteri,, cn most esily e verified y direct summtion of ll f(). Normlizing discrete PDF Discrete PDF s must stisfy. Sometimes, you hve the functionl form of the PDF nd you simply need to force it to stisfy this criterion. In tht cse you need to normlize the PDF so tht it sums to unity. If f () is n unnormlized PDF, then it cn e normlized y the multipliction of constnt, c (.) where tht constnt is the inverse of the sum of the unnormlized PDF. Emple..: Find the vlue of c tht normlizes the following PDF. ( ) f ) c 4 P ( for,,,, & 4 To normlize, we sum the PDF over ll vlues nd set it to unity. 4 f () + f () + f () + f () + f (4) 4 4 ( P ) c ( P ) c( P + P + P + P + P ) c( ) c We then solve for simplify nd solve for the normliztion constnt, c. c( 6) c 6 So the normlized PDF is

5 Rndom Vriles nd Proility Distriutions- 6 ( ) 4 P Discrete Cumultive Distriution Function (CDF) The discrete cumultive distriution function (CDF), F() of discrete rndom vrile X with the proility distriution, f(), is given y F( ) P( ) for - (.) The CDF is the proility tht is less thn or equl to. Emple.6.: In the ove emple, regrding the consumer purchsing devices, we cn otin the cumultive distriution directly: F() f() /8, F() f()+f()/8, F()f()+f()+f() Note: The cumultive distriution is lwys monotoniclly incresing, with. The finl vlue of the cumultive distriution is lwys unity, since the PDF is normlized. Proility Histogrm: A proility histogrm is grphicl representtion of the distriution of discrete rndom vrile. The histogrm for the PDF nd CDF for the Emple.4. re given in Figure.. The histogrm of the PDF provides visul representtion of the proility distriution, its most likely outcome nd the shpe of the distriution. The histogrm of the CDF provides visul representtion of the cumultive proility of n outcome. We oserve tht the CDF is monotoniclly incresing nd ends t one, s it must since the PDF is normlized nd sums to unity. Figure.. The histogrm of the PDF (top) nd CDF (ottom) for Emple.4.

6 Rndom Vriles nd Proility Distriutions -.4. Continuous Proility Density Functions (PDFs) Proility distriution functions of discrete rndom vriles re clled proility density functions when pplied to continuous vriles. Both hve the sme mening nd cn e revited commonly s PDF s. Proility density functions stisfy three criteri, which re nlogous to those for discrete PDFs, nmely for ll R d P( ) d (.4) The proility of finding n ect point on continuous rndom vrile is zero, P ( ) P( ) d Consequently, the proility tht rndom vrile is greter thn or greter thn or equl to numer is the sme in for continuous rndom vriles. The sme is true of less thn nd less thn or equl to signs for continuous rndom vriles. This equivlence is solutely not true for discrete rndom vriles. P( ) P( ) nd P( > > ) P( ) Also it is importnt to note tht sustitution of vlue into the PDF gives proility only for discrete rndom vrile, in order words P ( ) f ( ) for discrete PDFs only. For continuous rndom vrile, f () y itself doesn t provide proility. Only the integrl of f () provides proility from continuous rndom vrile. Emple.7.: A proility density function hs the form for - otherwise A plot of the proility density distriution is shown in Figure.. This plot is the continuous nlog of the discrete histogrm.

7 Rndom Vriles nd Proility Distriutions- Figure.. A plot of the PDF (left) nd CDF (right) for Emple.7. The proility of finding n etween nd is y (eqution.4) otherwise for - ) ( ) ( d d d f P otherwise for 9 9 ) ( ) ( d f P () Find ) ( P 9 ) ( 9 ) ( ) ( d f P This result mkes sense since the PDF is normlized nd we hve integrted over the entirety of the non-zero rnge of the rndom vrile. () Find ) ( P We cnnot integrte over discontinuities in function. Therefore, we must rek-up the integrl over continuous prts.

8 Rndom Vriles nd Proility Distriutions - P ( ) d + d + ( ) f d + 9 ( ) 9 + Here we see tht ctully it is not prcticlly necessry to integrte over the prts of the function where f(), ecuse the integrl over those rnges is lso. In generl prctice, we just need to perform the integrtion over those rnges where the PDF, f(), is non-zero. (c) Find P ( ) P ( ) d + d + 9 ( ) 9 9 Agin, it is not necessry to eplicitly integrte over nything ut the non-zero portions of the PDF, s ll other portions contriute nothing to the integrl. (d) Find P ( ) P ( ) d Testing continuous PDF for legitimcy If you re sked to determine if given PDF is legitimte, you re required to verify the three criteri in eqution (.4). Generlly, the third criterion is given in the prolem sttement, so you only hve to check the first criteri. The first criteri,, cn most esily e verified y plotting f() nd showing tht it is never negtive. The second criteri, d, cn most esily e verified y direct integrtion of f(). Normlizing continuous PDF Continuous PDF s must stisfy d. Sometimes, you hve the functionl form of the PDF nd you simply need to force it to stisfy this criterion. In tht cse you need to normlize the PDF so tht it sums to unity. If f () is n unnormlized PDF, then it cn e normlized y the multipliction of constnt,

9 Rndom Vriles nd Proility Distriutions- 4 c (.) f ( ) d where tht constnt is the inverse of the sum of the unnormlized PDF. Emple.8.: Find the vlue of c tht normlizes the PDF. c for - otherwise To normlize: 9 d c d c c c c So the normlized PDF is for - otherwise Continuous Cumultive distriutions The cumultive distriution F() of continuous rndom vrile with density function f() is F( ) P( ) d for (.6) This function gives the proility tht rndomly selected vlue of the vrile is less thn. The implicit lower limit of cumultive distriution is negtive infinity. F( ) P( ) P( ) Emple.9.: Determine the cumultive distriution function for the PDF of Emple.7.

10 Rndom Vriles nd Proility Distriutions - F( ) P( ) f ( t) dt for - for - for > A plot of the cumultive distriution function for the PDF of Emple.7. is shown in Figure.. The CDF is gin monotoniclly incresing. It egins t zero nd ends t unity, since the PDF is normlized... Reltions etween Inequlities In the ove section we hve defined specific function for the proility tht is less thn or equl to, nmely the cumultive distriution. But wht out when is greter thn, or strictly less thn, etc.? Here, we discuss those possiilities. Consider the fct tht the proility of ll outcomes must sum to one. Then we cn write (regrdless of whether the PDF is discrete or continuous) P ( ) + P( ) + P( > ) Using the union rule we cn write: P ( ) P[( ) ( )] P( ) + P( ) + P[( ) ( )] The intersection is zero, ecuse cnnot equl nd e less thn, so P( ) P[( ) ( )] P( ) + P( ) Similrly P( ) P[( > ) ( )] P( > ) + P( ) Using these three rules, we cn crete generlized method for otining ny ritrry proility. On the other hnd, we cn use the rules to crete wy to otin ny proility from just the cumultive distriution function. (This will e importnt lter when we use PDF s for which only the cumultive distriution function is given.) Regrdless of which method you use, you will otin the sme nswer. In Tle., we summrize the epression of ech proility in terms of the cumultive PDF. The continuous cse hs one importnt difference. In the continuous cse, the proility of rndom vrile equling single vlue is zero. Why? Becuse the proility is rtio of the numer of wys of getting over the totl numer of wys in the smple spce. There is only one

11 Rndom Vriles nd Proility Distriutions- 6 wy to get, nmely. But in the denomintor, there is n infinite numer of vlues of, since is continuous. Therefore, the P(). We cn show this using the definition is we write, P ( ) d for continuous PDF s only. One consequence of this is tht P ( ) P( ) + P( ) P( ) P ( ) P( > ) + P( ) P( > ) The proility of is the sme s. Likewise, the proility of is the sme s >. This fct mkes the continuous cse esy to generte. In Tle., we summrize the epression of ech proility in terms of the cumultive PDF. Proility Definition from cumultive PDF P ( ) f () P ( ) - P( -) P ( ) P ( ) P ( ) P ( ) P ( > ) > ( P ( -) f ) ( - P ( -) f ) ( P ( ) f ) Tle.. Reltions etween inequlities for discrete rndom vriles. Proility Definition from cumultive PDF P ( ) d P( ) or P ( ) P ( ) or P ( > ) d d P ( ) - P ( ) Tle.. Reltions etween inequlities for continuous rndom vriles.

12 Rndom Vriles nd Proility Distriutions - 7 Let s close out this section with two more voculry words used to descrie PDFs. The definitions of symmetric nd skewed distriutions re provided elow. An emple of ech re plotted in Figure.. Symmetric A proility density distriution is sid to e symmetric if it cn e folded long verticl is so tht the two sides coincide. Skew A proility density distriution is sid to e skewed if it is not symmetric. Figure.. Emples of symmetric nd skewed PDFs..6. Discrete Joint Proility Distriution Functions Thus fr in this chpter, we hve ssumed tht we hve only one rndom vrile. In mny prcticl pplictions there re more thn one rndom vrile. The ehvior of sets of rndom vriles is descried y Joint PDFs. In this ook, we eplicitly etend the formlism to two rndom vriles. It cn e etended to n ritrry numer of rndom vriles. We will present this etension twice, once for discrete rndom vriles nd once for continuous rndom vriles. The function f(, is joint proility distriution or proility mss function of the discrete rndom vrile X nd Y if f (, y f (, P( y ) f (, ) (.7)

13 Rndom Vriles nd Proility Distriutions- 8 This is just the two vrile etension of eqution (.). The PDF is lwys positive. The PDF is normlized, summing to unity, over ll comintions of nd y. The Joint PDF gives the intersection of the proility. The etension of the cumultive discrete proility distriution, eqution (.4), is tht for ny region A in the -y plne, F (, ) P( y ) f (, (.8) y Tht is to sy, the proility tht result (, is inside n ritrry re, A, is equl to the sum of the proilities for ll of the discrete events inside A. Emple..: Consider the discrete Joint PDF, f(,, s given in the tle elow. y / / / 4/ / / / / / Compute the proility tht is nd y is. P( y ) f (,) Compute the proility tht is less thn or equl to nd y is less thn or equl to. P( y ) y f (, f (,) + f (,) + f (,) Continuous Joint Proility Density Functions The distriution of continuous vriles cn e etended in n ectly nlogous mnner s ws done in the discrete cse. The function f(, is Joint Density Function of the continuous rndom vriles, nd y, if

14 Rndom Vriles nd Proility Distriutions - 9 f (, f (, ddy P[(, A] for ll, y A R f (, ddy (.9) This is just the two vrile etension of eqution (.4). The PDF is lwys positive. The PDF is normlized, integrting to unity, over ll comintions of nd y. The Joint PDF gives the intersection of the proility. Tht third eqution tkes specific form, depending on the shpe of the Are A. For rectngle, it would look like: d P ( c y d) f (, ddy c Nturlly, the cumultive distriution of the single vrile cse cn lso e etended to - vriles. F, P( y ) f (, ddy (.) ( Emple..: Given the continuous Joint PDF, find P(.. y ) f (, ( + for, y otherwise P( y ). 6y +. dy. ( + ddy y y y + dy +. 4 At this point, we should point out two things. First, we hve presented four cses for discrete nd continuous PDFs for one or two rndom vriles. There re relly only two core equtions, the requirements for the proility distriution nd the definition of the cumultive proility distriution. We hve shown these equtions for 4 cses; (i) discrete, one vrile, (ii) continuous one vrile, (iii) discrete, two vrile, nd (iv) continuous vrile. Re-emine these eight equtions to mke sure tht you see the similrities.

15 Rndom Vriles nd Proility Distriutions- 4 In this tet, we re stopping t two vriles. However, discrete nd continuous proility distriutions cn e functions of n ritrry numer of vriles..8. Mrginl Distriutions nd Conditionl Proilities Mrginl distriutions give us the proility of otining one vrile outcome regrdless of the vlue of the other vrile. Mrginl distriutions re needed to clculte conditionl proilities. The mrginl distriutions of lone nd of y lone re g ( ) f (, nd h( f (, (.) for the discrete cse nd y g ( ) f (, dy nd h( f (, d (.) for the continuous cse. Emple..: The discrete joint density function is given y the following tle. y / / / 4/ / / / / / Compute the mrginl distriution of t ll possile vlues of : g() f(,) + f(,) + f(,) 6/ g() f(,) + f(,) + f(,) 7/ g() f(,) + f(,) + f(,) 7/ Compute the mrginl distriution of y t ll possile vlues of y: h( f(,) + f(,) + f(,) 7/ h( f(,) + f(,) + f(,) / h(y) f(,) + f(,) + f(,) 8/ We note tht oth mrginl distriutions re legitimte PDFs nd stisfy the three requirements of eqution (.), nmely tht they re non-negtive, normlized nd their evlution yields proilities.

16 Rndom Vriles nd Proility Distriutions - 4 Emple..: The continuous joint density function is f (, ( + for, y otherwise Find g() nd h( for this joint density function. g( ) f (, dy dy + ( + dy + dy y + y ( + y) 4 + h( f (, d d + ( + d + d + 6y These mrginl distriutions themselves stisfy ll the properties of proility density distriution, nmely the requirements in eqution (.4). The physicl mening of the mrginl distriution functions re tht they give the individul effects of nd y seprtely. Conditionl Proility We now relte the conditionl proility to the mrginl distriutions defined ove. We do this first for the discrete cse nd then for the continuous cse. Let nd y e two discrete rndom vriles. The conditionl distriution of the rndom vrile y, given tht, is f (, y ) f ( y ) where g() > (.) g( ) Similrly, the conditionl distriution of the rndom vrile, given tht y, is f (, y ) f ( y ) where h() > (.4) h( y ) You should see tht this conditionl distriution is simply the ppliction of the definition of the conditionl proility, which we lerned in Chpter,

17 Rndom Vriles nd Proility Distriutions- 4 ( B A) ( A B) P( A) P P for P(A) > (.8) Emple.4.: Given the discrete PDF in Emple.., clculte () f ( y ) () f ( y ) () f ( y ). Using the conditionl proility definition: f ( y ) f (, y ) g( ) We lredy hve the denomintor: g() 7/. The numertor is f(, /. Therefore, the conditionl proility is: / f ( y ) 7 / () f ( y ) 7 f ( y ) f (, y ) h( y ) The numertor is the sum over ll vlues of f(, for which, nd y. So 4 f (, y ) f (,) + f (,) + f (,) The denomintor is the sum over ll h( for y 7 h ( y ) h() + h() + Therefore, 7/ f ( y ) / 7

18 Rndom Vriles nd Proility Distriutions - 4 A similr tretment cn e done for the continuous cse. Let nd y e two continuous vriles. The conditionl distriution of the rndom vrile cyd, given tht, is d f(,ddy c P( c y d ) where g()d g()d > (.) Similrly, the conditionl distriution of the rndom vrile, given tht cyd, is d f(,ddy d c P( c y d) where h(dy d c h(dy c > (.6) Emple..: Consider the continuous joint PDF in prolem.. Clculte P( X.. y ). P( X.. y ) P( X.. y ) P( X. ). f(,ddy. P( X.. y ) h(dy. We clculted the numertor in Emple.. nd it hd numericl vlue of /4. The denomintor is:. 6y y 6y h(dy + dy +.. The conditionl proility is then

19 Rndom Vriles nd Proility Distriutions- 44 P( X.. y ) 4.9. Sttisticl Independence 6 In Chpter, we used the conditionl proility rule to s check for independence of two outcomes. This sme pproch is repeted here for two rndom vriles. Let nd y e two rndom vriles, discrete or continuous, with joint proility distriution f(, nd mrginl distriutions g() nd h(. The rndom vriles nd y re sid to e sttisticlly independent iff (if nd only if) f (, g( ) h( if nd only if nd y re independent (.7) for ll possile vlues of (,. This should e compred with the rule for independence of proilities: P ( A B) P( A) P( B) iff nd A nd B re independent events (.44) Emple.6.: In the continuous emple given ove, determine whether nd y re sttisticlly independent rndom vriles. f (, ( + for, y otherwise 4 g ( ) + nd h( + 6y 4 6y g ( ) h( + + (8 + 4y The product of mrginl distriutions is not equl to the joint proility density distriution. Therefore, the vriles re not sttisticlly independent.

20 Rndom Vriles nd Proility Distriutions Prolems Prolem.. Determine the vlue of c so tht the following functions cn serve s PDF of the discrete rndom vrile X. ( 4) c + where,,,; Prolem.. A shipment of 7 computer monitors contins defective monitors. A usiness mkes rndom purchse of monitors. If is the numer of defective monitors purchsed y the compny, find the proility distriution of X. (This mens you need three numers, f(), f(), nd f() ecuse the rndom vrile, X numer of defective monitors purchsed, hs rnge from to. Also, find the cumultive PDF, F(). Plot the PDF nd the cumultive PDF. These two plots must e turned into clss on the dy the homework is due. Prolem.. A continuous rndom vrile, X, tht cn ssume vlues etween nd hs PDF given y + 7 ( ) Find () P(X4) nd find () P(X4). Plot the PDF nd the cumultive PDF. Prolem.4. Consider system of prticles tht sit in n electric field where the energy of interction with the electric field is given y E() , where is sptil position of the prticles. The proility distriution of the prticles is given y sttisticl mechnics to e f() c*ep(-e()/(r*t)) for nd otherwise, where R 8.4 J/mol/K nd T 7. Kelvin. () Find the vlue of c tht mkes this legitimte PDF. () Find the proility tht prticles sits t. (c) Find the proility tht prticles sits t >.7 (d) Find the proility tht prticles sits t..7 Prolem.. Let X denote the rection time, in seconds, to certin stimulnt nd Y denote the temperture (reduced units) t which certin rection strts to tke plce. Suppose tht the rndom vriles X nd Y hve the joint PDF,

21 cy for ; y. f (, elsewhere where c.979. Find () ( X nd Y ) Rndom Vriles nd Proility Distriutions- 46 P nd () ( X Y ) 4 P. Prolem.6. Let X denote the numer of times tht control mchine mlfunctions per dy (choices:,, ) nd Y denote the numer of times technicin is clled. f(, is given in tulr form. f(,... y () Evlute the mrginl distriution of X. () Evlute the mrginl distriution of Y. (c) Find P(Y X ).