Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts into mthemticl frmework. We invoke lger for discrete vriles nd clculus for continuous vriles. Ever topic in this chpter is presented twice, once for discrete vriles nd gin for continuous vriles. The nlog in the two cses should e pprent nd should reinforce the common underling 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 proilit to tht of mthemtics cn e performed not onl when we hve single vrile of interest, ut lso when we hve two vriles. Agin, high-lighting this nlog etween single nd joint proilit distriutions eplicitl revels the common underling concepts... Rndom Vriles & Smple Spces We egin with the introduction of necessr voculr. 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 mn 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.
Rndom Vriles nd Proilit Distriutions - 7 result 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 Proilit Distriution Functions PDFs Proilit distriution function PDF The function, f is proilit distriution function of the discrete rndom vrile, if for ech possile outcome, the following three criteri re stisfied. f f. P f The PDF is lws non-negtive. The PDF is normlized, mening tht the sum over ll vlues of discrete PDF is unit. The PDF evluted t outcome provides the proilit of the occurrence of outcome. Emple.4.: Eight devices re shipped to retil outlet, of which re defective. If consumer purchses computers, find the proilit distriution for the numer of defective devices ought 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 the
Rndom Vriles nd Proilit Distriutions- 8 consumer. The rndom vrile,, cn tke on vlues of,, nd. Those re the onl numer of defective devices the consumer cn u, given tht the re onl uing two devices. The net step is to determine the size of the smple spce. The numer of ws 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 u. Third, the proilit of prticulr outcome is equl to the numer of ws to get tht outcome over the totl numer of ws: f P ws of getting totl ws f P 8 f P 8 f P 8 8 8 8 In ech of these cses, we otined the numertor, the numer of ws of getting outcome, using the comintion rule nd the generlized multipliction rule. There re ws of choosing defective devices from defective devices. There re ws of choosing - - good devices from good devices. We use the generlized multipliction rule to get the numer of ws of getting oth of these outcomes in the numertor. As preview, we will come to discover tht this proilit distriution is clled the hpergeometric 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 proilit for getting or n numer other thn,, or defective devices, then the proilit is zero nd f=. Testing discrete PDF for legitimc If ou re sked to determine if given PDF is legitimte, ou re required to verif the three criteri in eqution.. Generll, the third criterion is given in the prolem sttement, so ou onl hve to check the first two criteri.
Rndom Vriles nd Proilit Distriutions - 9 The first criteri, f, cn most esil e verified plotting f nd showing tht it is never negtive. The second criteri, f, cn most esil e verified direct summtion of ll f. Normlizing discrete PDF Discrete PDF s must stisf f. Sometimes, ou hve the functionl form of the PDF nd ou simpl need to force it to stisf this criterion. In tht cse ou need to normlize the PDF so tht it sums to unit. If f is n unnormlized PDF, then it cn e normlized the multipliction of constnt, f cf f. f 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 unit. f 4 f f f f f f 4 4 4 P c P c P P P P P c 4 4 4 f c 4 4 4 4 4 4 4 4 We then solve for simplif nd solve for the normliztion constnt, c. c 6 c 6 So the normlized PDF is
Rndom Vriles nd Proilit Distriutions- 6 f 4 P Discrete Cumultive Distriution Function CDF The discrete cumultive distriution function CDF, F of discrete rndom vrile X with the proilit distriution, f, is given F P f for -. The CDF is the proilit tht is less thn or equl to. Emple.6.: In the ove emple, regrding the consumer purchsing devices, we cn otin the cumultive distriution directl: F = f = /8, F = f+f=/8, F=f+f+f= Note: The cumultive distriution is lws monotonicll incresing, with. The finl vlue of the cumultive distriution is lws unit, since the PDF is normlized. Proilit Histogrm: A proilit 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 proilit distriution, its most likel outcome nd the shpe of the distriution. The histogrm of the CDF provides visul representtion of the cumultive proilit of n outcome. We oserve tht the CDF is monotonicll incresing nd ends t one, s it must since the PDF is normlized nd sums to unit. Figure.. The histogrm of the PDF top nd CDF ottom for Emple.4.
Rndom Vriles nd Proilit Distriutions -.4. Continuous Proilit Densit Functions PDFs Proilit distriution functions of discrete rndom vriles re clled proilit densit functions when pplied to continuous vriles. Both hve the sme mening nd cn e revited commonl s PDF s. Proilit densit functions stisf three criteri, which re nlogous to those for discrete PDFs, nmel f for ll R f d P f d.4 The proilit of finding n ect point on continuous rndom vrile is zero, P P f d Consequentl, the proilit 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 solutel 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 proilit onl for discrete rndom vrile, in order words P f for discrete PDFs onl. For continuous rndom vrile, f itself doesn t provide proilit. Onl the integrl of f provides proilit from continuous rndom vrile. Emple.7.: A proilit densit function hs the form f for - otherwise A plot of the proilit densit distriution is shown in Figure.. This plot is the continuous nlog of the discrete histogrm.
Rndom Vriles nd Proilit Distriutions- Figure.. A plot of the PDF left nd CDF right for Emple.7. The proilit of finding n etween nd is 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 entiret 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.
Rndom Vriles nd Proilit Distriutions - P f d f d f d 9 9 Here we see tht ctull it is not prcticll necessr 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 f d f d 9 9 9 Agin, it is not necessr to eplicitl integrte over nthing ut the non-zero portions of the PDF, s ll other portions contriute nothing to the integrl. d Find P P f d 9 9 9 Testing continuous PDF for legitimc If ou re sked to determine if given PDF is legitimte, ou re required to verif the three criteri in eqution.4. Generll, the third criterion is given in the prolem sttement, so ou onl hve to check the first criteri. The first criteri, f, cn most esil e verified plotting f nd showing tht it is never negtive. The second criteri, f d, cn most esil e verified direct integrtion of f. Normlizing continuous PDF Continuous PDF s must stisf f d. Sometimes, ou hve the functionl form of the PDF nd ou simpl need to force it to stisf this criterion. In tht cse ou need to normlize the PDF so tht it sums to unit. If f is n unnormlized PDF, then it cn e normlized the multipliction of constnt,
Rndom Vriles nd Proilit Distriutions- 4 f cf f. 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 f for - otherwise To normlize: 9 f d c d c c c c So the normlized PDF is f for - otherwise Continuous Cumultive distriutions The cumultive distriution F of continuous rndom vrile with densit function f is F P f d for -.6 This function gives the proilit tht rndoml selected vlue of the vrile is less thn. The implicit lower limit of cumultive distriution is negtive infinit. F P P Emple.9.: Determine the cumultive distriution function for the PDF of Emple.7.
Rndom Vriles nd Proilit Distriutions - F P f t dt 9 9 for - for - for A plot of the cumultive distriution function for the PDF of Emple.7. is shown in Figure.. The CDF is gin monotonicll incresing. It egins t zero nd ends t unit, since the PDF is normlized... Reltions etween Inequlities In the ove section we hve defined specific function for the proilit tht is less thn or equl to, nmel the cumultive distriution. But wht out when is greter thn, or strictl less thn, etc.? Here, we discuss those possiilities. Consider the fct tht the proilit 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 Similrl P P[ ] P P Using these three rules, we cn crete generlized method for otining n ritrr proilit. On the other hnd, we cn use the rules to crete w to otin n proilit from just the cumultive distriution function. This will e importnt lter when we use PDF s for which onl the cumultive distriution function is given. Regrdless of which method ou use, ou will otin the sme nswer. In Tle., we summrize the epression of ech proilit in terms of the cumultive PDF. The continuous cse hs one importnt difference. In the continuous cse, the proilit of rndom vrile equling single vlue is zero. Wh? Becuse the proilit is rtio of the numer of ws of getting over the totl numer of ws in the smple spce. There is onl one
Rndom Vriles nd Proilit Distriutions- 6 w to get, nmel =. 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 f d for continuous PDF s onl. One consequence of this is tht P P P P P P P P The proilit of is the sme s. Likewise, the proilit of is the sme s. This fct mkes the continuous cse es to generte. In Tle., we summrize the epression of ech proilit in terms of the cumultive PDF. Proilit Definition from cumultive PDF P f P - P - P f P P f P - P f P - P f P Tle.. Reltions etween inequlities for discrete rndom vriles. Proilit Definition from cumultive PDF P f d P or P P or P f d f d P - P Tle.. Reltions etween inequlities for continuous rndom vriles.
Rndom Vriles nd Proilit Distriutions - 7 Let s close out this section with two more voculr words used to descrie PDFs. The definitions of smmetric nd skewed distriutions re provided elow. An emple of ech re plotted in Figure.. Smmetric A proilit densit distriution is sid to e smmetric if it cn e folded long verticl is so tht the two sides coincide. Skew A proilit densit distriution is sid to e skewed if it is not smmetric. Figure.. Emples of smmetric nd skewed PDFs..6. Discrete Joint Proilit Distriution Functions Thus fr in this chpter, we hve ssumed tht we hve onl one rndom vrile. In mn prcticl pplictions there re more thn one rndom vrile. The ehvior of sets of rndom vriles is descried Joint PDFs. In this ook, we eplicitl etend the formlism to two rndom vriles. It cn e etended to n ritrr 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 proilit distriution or proilit mss function of the discrete rndom vrile X nd Y if f, f, P f,.7
Rndom Vriles nd Proilit Distriutions- 8 This is just the two vrile etension of eqution.. The PDF is lws positive. The PDF is normlized, summing to unit, over ll comintions of nd. The Joint PDF gives the intersection of the proilit. The etension of the cumultive discrete proilit distriution, eqution.4, is tht for n region A in the - plne, F, P f,.8 Tht is to s, the proilit tht result, is inside n ritrr 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. / / / 4/ / / / / / Compute the proilit tht is nd is. P f, Compute the proilit tht is less thn or equl to nd is less thn or equl to. P f, f, f, f, 4 7.7. Continuous Joint Proilit Densit Functions The distriution of continuous vriles cn e etended in n ectl nlogous mnner s ws done in the discrete cse. The function f, is Joint Densit Function of the continuous rndom vriles, nd, if
Rndom Vriles nd Proilit Distriutions - 9 f, f, dd P[, A] for ll, A R f, dd.9 This is just the two vrile etension of eqution.4. The PDF is lws positive. The PDF is normlized, integrting to unit, over ll comintions of nd. The Joint PDF gives the intersection of the proilit. Tht third eqution tkes specific form, depending on the shpe of the Are A. For rectngle, it would look like: d P c d f, dd c Nturll, the cumultive distriution of the single vrile cse cn lso e etended to - vriles. F, P f, dd. Emple..: Given the continuous Joint PDF, find P.. f, for, otherwise P. 6. d. dd d. 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 rell onl two core equtions, the requirements for the proilit distriution nd the definition of the cumultive proilit 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 ou see the similrities.
Rndom Vriles nd Proilit Distriutions- 4 In this tet, we re stopping t two vriles. However, discrete nd continuous proilit distriutions cn e functions of n ritrr numer of vriles..8. Mrginl Distriutions nd Conditionl Proilities Mrginl distriutions give us the proilit 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 lone re g f, nd h f,. for the discrete cse nd g f, d nd h f, d. for the continuous cse. Emple..: The discrete joint densit function is given the following tle. / / / 4/ / / / / / Compute the mrginl distriution of t ll possile vlues of : g= = f, + f, + f, = 7/ g= = f, + f, + f, = / g= = f, + f, + f, = 8/ Compute the mrginl distriution of t ll possile vlues of : h= = f, + f, + f, = 6/ h= = f, + f, + f, = 7/ h= = f, + f, + f, = 7/
Rndom Vriles nd Proilit Distriutions - 4 We note tht oth mrginl distriutions re legitimte PDFs nd stisf the three requirements of eqution., nmel tht the re non-negtive, normlized nd their evlution ields proilities. Emple..: The continuous joint densit function is otherwise, for, f Find g nd h for this joint densit function. 4, d d d d f g 6, d d d d f h These mrginl distriutions themselves stisf ll the properties of proilit densit distriution, nmel the requirements in eqution.4. The phsicl mening of the mrginl distriution functions re tht the give the individul effects of nd seprtel. Conditionl Proilit We now relte the conditionl proilit to the mrginl distriutions defined ove. We do this first for the discrete cse nd then for the continuous cse. Let nd e two discrete rndom vriles. The conditionl distriution of the rndom vrile =, given tht =, is where, g g f f. Similrl, the conditionl distriution of the rndom vrile =, given tht =, is h h f f where,.4
Rndom Vriles nd Proilit Distriutions- 4 You should see tht this conditionl distriution is simpl the ppliction of the definition of the conditionl proilit, which we lerned in Chpter, P B A P A B P A for PA.8 Emple.4.: Given the discrete PDF in Emple.., clculte f f f. Using the conditionl proilit definition: f f, g We lred hve the denomintor: g= = /. The numertor is f=,= = /. Therefore, the conditionl proilit is: / f / f f f, h The numertor is the sum over ll vlues of f, for which =, nd. So 4 f, f, f, f, 7 The denomintor is the sum over ll h for 6 7 7 h h h h Therefore,
Rndom Vriles nd Proilit Distriutions - 4 7/ f / 7 A similr tretment cn e done for the continuous cse. Let nd e two continuous vriles. The conditionl distriution of the rndom vrile c<<d, given tht <<, is P c d d c f,dd gd where gd. Similrl, the conditionl distriution of the rndom vrile <<, given tht c<<d, is P c d d c d c f,dd hd where d c hd.6 Emple..: Consider the continuous joint PDF in prolem.. Clculte P X... P X.. P X.. P X.. f,dd. P X.. hd. We clculted the numertor in Emple.. nd it hd numericl vlue of /4. The denomintor is:. 6 6 hd d.. The conditionl proilit is then
Rndom Vriles nd Proilit Distriutions- 44 P X.. 4.9. Sttisticl Independence 6 In Chpter, we used the conditionl proilit rule to s check for independence of two outcomes. This sme pproch is repeted here for two rndom vriles. Let nd e two rndom vriles, discrete or continuous, with joint proilit distriution f, nd mrginl distriutions g nd h. The rndom vriles nd re sid to e sttisticll independent iff if nd onl if f, g h if nd onl if nd 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 re sttisticll independent rndom vriles. f, for, otherwise 4 g nd h 6 4 6 g h 8 4 6 8 The product of mrginl distriutions is not equl to the joint proilit densit distriution. Therefore, the vriles re not sttisticll independent.
Rndom Vriles nd Proilit Distriutions - 4.. Prolems Prolem.. Determine the vlue of c so tht the following functions cn serve s PDF of the discrete rndom vrile X. f c 4 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 the compn, find the proilit distriution of X. This mens ou 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 d the homework is due. Prolem.. A continuous rndom vrile, X, tht cn ssume vlues etween = nd = hs PDF given f 7 Find PX<4 nd find P<X<4. Plot the PDF nd the cumultive PDF. Prolem.4. Consider sstem of prticles tht sit in n electric field where the energ of interction with the electric field is given E = 477.7 + 49.44, where is sptil position of the prticles. The proilit distriution of the prticles is given 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 proilit tht prticles sits t <. c Find the proilit tht prticles sits t >.7 d Find the proilit 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,
Rndom Vriles nd Proilit Distriutions- 46 c for f, ;. elsewhere where c =.979. Find X nd Y P nd X Y 4 P. Prolem.6. Let X denote the numer of times tht control mchine mlfunctions per d choices:,, nd Y denote the numer of times technicin is clled. f, is given in tulr form. f,......... Evlute the mrginl distriution of X. Evlute the mrginl distriution of Y. c Find PY = X =.