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Distribution Functions

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Disriuion Funions A disriuion funion is funion whih desries how vlues re lloed ross populion or smple spe. There re vriey of seings for whih suh funions rise nd we shll urn our enion o some emples. Emple
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Disriuion Funions A disriuion funion is funion whih desries how vlues re lloed ross populion or smple spe. There re vriey of seings for whih suh funions rise nd we shll urn our enion o some emples. Emple 1: Suppose you flip fir oin en imes. Wh is he disriuion of he numer of heds h you will oserve? Plo he disriuion. There re ol of 1024 differen ominions of heds nd ils h we n oserve sine we hve en slos o fill nd in eh slo, here re only wo possile vlues, Hed or Til. So, we hve 2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2 = 2 10 = 1024 differen ouomes. We ould oun eh of he 1,024 ouomes nd reord he numer of heds. This would ke lile while nd in he ineres of spe, he deils re omied. Insed, we end up wih he following le: Numer of Heds Proporion showing h mny heds We n lso disply our d in hisogrm. A hisogrm is grphil wy o represen d where veril rs re ped ove eh egory (group) in suh wy h he re of eh r represens he proporion of he populion in h egory. Figure 1: Hisogrm of Coin Flips 1 Noie h we grouped he d in only five egories. If we hd insed mde le for eh oin flip, we ould ge eer esimes. And we migh e ineresed in fiing smooh urve o he d. The urve h we fi should hve he propery h he re under he urve ove one group is equl o he re of he orresponding rengle. Doing his, we hve he following figure. Figure 2: Hisogrm of Coin Flips wih Smooh Curve The smooh funion h we pplied o he hisogrm ove is lled (proiliy) densiy funion. This funion hs he propery h Proporion of populion for Are under he grph of p( d ) whih is eween nd p( ) eween nd If nd re he smlles nd lrges vlues h our d n ke. (In his se, he smlles numer of heds we n oserve is 0 nd he lrges is 10), hen we hve h 10 p( d ) pd ( ) 1 0 So fr, we hve only onsidered he inegrl of p() nd no he funion p() iself. Looking Figure 2, noie h when = 5, urve is pproimely equl o This does no men h 0.25 of he oin flips will ome up wih 5 heds, in generl. Rher, we inerpre p(5) = 0.25 o men h for some ll inervl, D, round 5, he proporion of he smples wih numer of heds in his inervl is pproimely equl o p(5)d = 0.25D. 2 (Proiliy) densiy funions lso hve o hve wo oher imporn properies. Firs, hey mus e non-negive. This is euse he inegrl lwys gives proporion of he populion, whih iself mus e posiive numer eween 0 nd 1. Seond, he inegrl of p() from - o is equl o 1. This is euse in he inervl (-, ), we will hve oserved everyhing possile. Therefore, he proporion of populion will e equl o 1. We reord hese properies of (proiliy) densiy funion in he o elow. (Proiliy) Densiy Funion A funion p() is lled (proiliy) densiy funion if nd Proporion of populion for Are under he grph of p( d ) whih is eween nd p( ) eween nd p( d ) 1 nd 0 p() 1 for ll Emple 2: Suppose rel numer is hosen rndom on he inervl [1, 6]. Suh densiy funion is known s uniform disriuion, sine ll of he poins re eqully likely o e hosen. The grph of his funion is s follows: p Figure 3: A uniform disriuion on [1, 6] Find he vlue of whih will mke his densiy funion. Sine he re under he urve needs o equl 1 nd we re deling wih rengulr re wih se of 6 1 = 5 nd heigh of, we hve he equion 5 = 1. Solving for, we see h = 1/5. 3 Anoher wy of disussing how vlues re disriued is y using wh is known s umulive disriuion funion. I is defined s he funion P () pd ( ) Proporion of populion hving vlues of elow where p() is he (proiliy) densiy funion disussed ove. Noie h y he Fundmenl Theorem of Clulus, he funion P() is he niderivive of p(), wih lim P ( ) 0. We use he word umulive o denoe he f h his funion mesures he re under he urve from - o. Reurning o Emple 2, he shded region in he ne figure shows he proporion of numers h re eween 1 nd 3. p 0.2 Figure 4: Proporion of numers eween 1 nd 3 The umulive disriuion funion h orresponds o he shded res is given y he following grph: P Figure 5: Proporion of numers less hn In priulr, noie h he grph is onsn vlue of 1 fer = 6. This orresponds o he f h fer 6, we hve seen ll of he possile numers h we ould rndomly sele. We reord hese properies of umulive densiy funion in he o elow. 4 Cumulive Disriuion Funion A funion P() is lled umulive disriuion funion if P () pd ( ) And i sisfies he following properies 1. P() is non-deresing funion 2. l im P ( ) 1 nd lim P ( ) 0 Proporion of populion hving vlues of elow 3. Proporion of populion hving p( d ) P ( ) P ( ) vlues of eween nd Emple 3: Deermine if he following grphs orrespond o proiliy densiy funion or umulive disriuion funion nd find he vlue of. 2 4 The firs grph orresponds o umulive disriuion funion sine he grph rehes pleu nd remins here. Moreover, sine lim P ( ) 1, we see h = 1. The seond grph orresponds o proiliy densiy funion. Rell, he re under he urve mus e equl o 1. Here, we hve ringle wih se nd heigh 2. The re of he ringle is hus (1/2)ÿ2ÿ, so we see h = 1 s well. 5
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