Random Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois
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1 Radom Varables ECE 313 Probablty wth Egeerg Alcatos Lecture 8 Professor Rav K. Iyer Uversty of Illos Iyer - Lecture 8 ECE 313 Fall 013
2 Today s Tocs Revew o Radom Varables Cumulatve Dstrbuto Fucto (CDF Probablty Mass Fucto (PMF Dscrete Radom Varables Beroull Bomal Posso Geometrc Iyer - Lecture 8 ECE 313 Fall 013
3 Cumulatve Dstrbuto Fucto (CDF Some roertes of cdf are: F(b. lm b + F( b F( 1, lm F( b F( 0.. s a o-decreasg fucto of b,. b F a < b { a} Proerty ( follows sce for the evet s cotaed the evet, ad so t must have a smaller robablty. { b} Proertes ( ad ( follow sce must take o some fte value. All robablty questos about ca be aswered terms of cdf. For examle: F( P { a b} F( b F( a for all a < b.e. calculate b ( F( b P{ a b} by frst comutg the robablty that a ( F( a ad the subtract from ths the robablty that. Iyer - Lecture 8 ECE 313 Fall 013
4 Dscrete Radom Varables A radom varable that ca take o at most coutable umber of ossble values s sad to be dscrete. For a dscrete radom varable, we defe the robablty mass fucto of by: (a ( a P{ a} (a s ostve for at most a coutable umber of values of a..e., f Sce take values x: must assume oe of the values x1, x,, the ( x ( x > 0, 1,,... 0, for other values of x 1 ( x 1 Iyer - Lecture 8 ECE 313 Fall 013
5 Cumulatve Dstrbuto Fucto The cumulatve dstrbuto fucto ca be exressed terms of (a by: x Suose has a robablty mass fucto gve by F ( a ( all x a ( 1, (, (3 3 6 the the cumulatve dstrbuto fucto F of s gve by F( a F 0, 1 a < 1, 1 a < 5, a < 6 3 a 1 3 Iyer - Lecture 8 ECE 313 Fall 013
6 Dscrete Radom Varables Examles Beroull Bomal Posso Geometrc Iyer - Lecture 8 ECE 313 Fall 013
7 The Beroull Radom Varable (0 P{ P{ 0} 1 1}, Where, 0 1 s the robablty that the tral s a success s sad to be a Beroull radom varable f ts robablty mass fucto s gve by the above equato some for (0,1 Iyer - Lecture 8 ECE 313 Fall 013
8 The Bomal Radom Varable deedet trals, each of whch results a success wth ad a falure wth robablty 1- If reresets the umber of successes that occur the trals, s sad to be a bomal radom varable wth arameters (, The robablty mass fucto of a bomal radom varable havg arameters (, s gve by (, Equato (.3 where! (!! equals the umber of dfferet grous of objects that ca be chose from a set of objects Iyer - Lecture 8 ECE 313 Fall 013
9 The Bomal Radom Varable Equato (.3 may be verfed by frst otg that the robablty of ay artcular sequece of the outcomes cotag successes ad - falures s, by the assumed deedece of trals, Equato (.3 the follows sce there are dfferet sequeces of the outcomes leadg to I successes ad - 3 falures. For stace f 3,, the there are 3 ways whch the three trals ca result two successes. By the bomal theorem, the robabltes sum to oe: 0 ( 0 ( + 1 Iyer - Lecture 8 ECE 313 Fall 013
10 Bomal Radom Varable Examle 1 Four far cos are fled. Outcomes are assumed deedet, what s the robablty that two heads ad two tals are obtaed? Lettg equal the umber of heads ( successes that aear, the s a bomal radom varable wth arameters ( 4, 1. Hece by the bomal equato, P{ } 4 ( 1 1 ( 3 8 Iyer - Lecture 8 ECE 313 Fall 013
11 Bomal Radom Varable Examle It s kow that ay tem roduced by a certa mache wll be defectve wth robablty 0.1, deedetly of ay other tem. What s the robablty that a samle of three tems, at most oe wll be defectve? If s the umber of defectve tems the samle, the s a bomal radom varable wth arameters (3, 0.1. Hece, the desred robablty s gve by: P{ 0} + P{ 1} 3 ( ( ( ( Iyer - Lecture 8 ECE 313 Fall 013
12 Bomal RV Examle 3 Suose that a arlae ege wll fal, whe flght, wth robablty 1 deedetly from ege to ege; suose that the arlae wll make a successful flght f at least 50 ercet of ts eges rema oeratve. For what values of s a four-ege lae referable to a two-ege lae? Because each ege s assumed to fal or fucto deedetly of what haes wth the other eges, t follows that the umber of eges remag oeratve s a bomal radom varable. Hece, the robablty that a four-ege lae makes a successful flght s: Iyer - Lecture 8 ECE 313 Fall 013
13 Iyer - Lecture 8 ECE 313 Fall 013 Bomal RV Examle 3 (Cot The corresodg robablty for a two-ege lae s: The four-ege lae s safer f: Or equvaletly f: Hece, the four-ege lae s safer whe the ege success robablty s at least as large as /3, whereas the two-ege lae s safer whe ths robablty falls below / (3 1 ( or or
14 Iyer - Lecture 8 ECE 313 Fall 013 The Posso Radom Varable A radom varable, takg o oe of the values 0,1,,, s sad to be a Posso radom varable wth arameter, f for some >0, defes a robablty mass fucto sce 1! ( 0 0 e e e 0,1,...,! } { ( e P
15 Iyer - Lecture 8 ECE 313 Fall 013 The Posso Radom Varable Cot d A mortat roerty of the Posso radom varable, t may be used to aroxmate a bomal radom varable wth the bomal arameter s large ad s small Now, for large ad small Hece for large ad small P / /! 1 ( 1 ( 1!! (!!! (! } { +! } { e P 1 1-1, 1 ( 1 (, e
16 Cotuous Radom Varables Radom varables whose ossble varables whose set of ossble values s ucoutable s a cotuous radom varable f there exsts a oegatve fucto f(x defed for all real, havg the roerty that for ay set of B real umbers f(x s called the robablty desty fucto of the radom varable The robablty that wll be B may be obtaed by tegratg the robablty desty fucto over the set B. Sce must assume some value, f(x must satsfy 1 P { (, } f ( x dx Iyer - Lecture 8 ECE 313 Fall 013
17 Cotuous Radom Varables Cot d All robablty statemets about ca be aswered terms of f(x b e.g. lettg B[a,b], we obta P { a b} f ( x dx a If we let ab the recedg, the P{ a} f ( x dx 0 a a Ths equato states that the robablty that a cotuous radom varable wll assume ay artcular value s zero The relatosh betwee the cumulatve dstrbuto F( ad the robablty desty f( a F ( a P{ (, a} f ( x dx Dfferetatg both sdes of the recedg yelds d da F( a f ( a Iyer - Lecture 8 ECE 313 Fall 013
18 Cotuous Radom Varables Cot d That s, the desty of the dervatve of the cumulatve dstrbuto fucto. A somewhat more tutve terretato of the desty fucto whe ε s small ε P a ε a + a + ε / aε / f ( x dx εf ( a The robablty that wll be cotaed a terval of legth ε aroud the ot a s aroxmately εf(a Iyer - Lecture 8 ECE 313 Fall 013
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