Experece has show that a certa le detector wll show a postve readg (so you are lyg) 0% of the tme whe a perso s tellg the truth ad 95% of the tme whe a perso s actually lyg. Suppose 0 suspects are subjected to a le detector test regardg a recet oe-perso crme. The probablty of observg o postve readg f all suspects pleaded ocet ad are tellg the truth s? I some courses (but certaly ot a tro stats course!) studets are graded o a ormal curve. For example, studets wth 0.5 sd from the mea receve a C; betwee 0.5 ad stadard devatos above the mea receve a C+ betwee ad.5 stadard devatos above the mea receve a B betwee.5 ad stadard devatos above the mea receve a B+ The class mea value a exam was 60 wth a stadard devato of 0. Assume the grades are dstrbuted ormally. What are the bouds for a B grade? What percetage of the studet wll receve a B?
Lecture 4: Dstrbuto of the Mea of Radom Varables Law of large umbers Cetral lmt theorem Mea of Radom Varables,, 3,, all come from the same dstrbuto wth expected value ad varace. Assume that they are also depedet of each other. The we ca defe a ew radom varable whch s the mea of these r.v.s: The mea of radom varables has a few trats that are credble ad later the course wll help us a lot to coect probablty wth statstcs
Mea of Radom Varables Example: Let ad be two depedet tosses of a far co where Heads has a value of ad Tal a value of 0. What s the dstrbuto of the mea? Example: Now what happes whe we crease the umber of radom varables (). What s the dstrbuto of the mea? The law of large umber As creases, the mea of radom varables wll coverge to the expected value Does ths make sese to you? Why t should ot make sese: a radom varable coverges to a costat!!!!!! Why t should make sese: The more values you see from the dstrbuto the more formato you have about the expected value (the mddle of the dstrbuto) 3
4 The dstrbuto of a mea of radom varables Frst, we exame the ceter of ts dstrbuto. The expected value of the mea s What s the expected value of the mea of two co tosses? Ad for three co tosses? E E E ) ( ) ( The dstrbuto of a mea of radom varables Now we exame the spread of ts dstrbuto. The varace of the mea of radom varables s As creases, varace of the mea decreases! The Law of Large Number should make better sese ow, why? What s the varace of the mea of two co tosses? Var Var Var ) ( ) (
Dstrbuto of Mea of radom varables We ow kow the ceter ad spread of the mea of radom varables. What about the shape of the dstrbuto? If,,, follow a Normal dstrbuto, the ther mea wll also follow a Normal dstrbuto! Example: Dstrbuto for mea of Normals R.V. 5
Cetral Lmt Theorem What f the radom varables do ot follow a Normal dstrbuto? Lets start wth a radom varable that has the followg dstrbuto: Cetral Lmt Theorem 6
Cetral Lmt Theorem If the s large eough, the the mea has a approxmately Normal dstrbuto wth expected value ad varace Ths s true o matter what the shape of the dstrbuto of the orgal populato! Ths s remarkable (f ths makes perfect sese to you the you should jo the statstcs departmet ). Cetral Lmt Theorem for Totals! If the s large eough, the the total W has a approxmately Normal dstrbuto wth expected value ad varace Where E( Var( ) ) 7
Cetral Lmt Theorem for Totals! A group of 5 people wat to get to the elevator. There s a weght lmt of 500 pouds. Let be the weght of perso gog o the elevator. The mea value of weght s 70 pouds ad the SD s 0 pouds. What s the probablty that the group total weght wll exceed the maxmum value? W- total weght of the 5 people =5,E()=70,SD()=0 E(W)=5*70=550 Var(W)=5*0^=6000 SD(W)=464.758 500 550 P( W 500) P( Z ) P( Z -0.075) P( Z 0.075) 0.54. 464.758 Next Class - Lecture 5 Dscrete data: dstrbuto for proportos!!!! Ad the coecto to bomal ad ormal! 8
The dstrbuto of a mea of radom varables What exactly do we mea by the dstrbuto of the mea of radom varables? Sample from,,, Sample from,,, Sample from,,, Calculate the mea Calculate the mea Calculate the mea 3 Sample from,,, Calculate the mea m 9