7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f (x) Shows the probablty desty for all values of x. A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Probablty desty s ot probablty! Use a mathematcal model to descrbe the varable. Test scores 3 Cotuous Dstrbuto Probablty Desty Fucto (p.d.f.) of a radom varable of cotuous type wth a space S s a tegrable fucto,, that satsfyg the followg codtos:. f (x), x S,. S dx =, (Total area uder curve s.) b 3. For a ad b S, P ( a b) f ( x) dx a a b x 4 Meag of Area Uder Curve Example: What percetage of the dstrbuto s betwee 7 ad 86? The P(7 86) The 7 86 (Heght) P( = 7) =, desty s ot probablty. 5 6 ND -
7//3 Probablty Desty Fucto The cotuous radom varable has a ormal dstrbuto f ts p.d.f. s The mea, varace, ad m.g.f. of a cotuous radom varable that has a ormal dstrbuto are: f ( x) e x- - - < x < E[ ], M ( t) e t t / Var[ ], Notato: N (, ) A ormal dstrbuto wth mea ad stadard devato 7 8 m.g.f. of Example: If the m.g.f. of a radom varable s M(t) = exp(t + 6t ), what s the dstrbuto of ths radom varable? What are the mea ad stadard devato of ths dstrbuto ad what s the p.d.f. of ths radom varable? Dstrbuto? Mea? Stadard devato? p.d.f.? 9. Bell-Shaped & Symmetrcal. Mea, Meda, Mode Are Equal 3. Radom Varable Has Ifte Rage - < x < f() Mea Meda Mode Example Effect of Varyg Parameters ( & ) N (7, 5) A ormal dstrbuto wth mea 7 ad varace 5. Possble stuatos: Test scores, pulse rates, B N (3, 4) A ormal dstrbuto wth mea 3 ad varace 4. Possble stuatos: Weght, Cholesterol levels, A C x ND -
7//3 Probablty Ifte Number of Tables Probablty s area uder curve! P( c x d) f ( x) dx c d dstrbutos dffer by mea & stadard devato. f() Each dstrbuto would requre ts ow table. c d x That s a fte umber! 3 4 Stadard Stadard : A ormal dstrbuto wth mea = ad stadard devato =. Area uder Stadard Curve Notato: ~ N ( =, = ) Cap letter = 5 z How to fd the proporto of the are uder the stadard ormal curve below z or say P ( < z ) =? Use Stadard Table!!! 6 Stadard P( <.3) = F(.3) Area below.3 =.655?.3 7 8 ND - 3
7//3 Stadard P( >.3) = Area above.3 = -.655 =.3745 Areas the upper tal of the stadard ormal dstrbuto Stadard P( < <.3) = Area betwee ad.3 =? =.655.5 =.55.3.3 9 P ( -. < <. ) =?.686 P ( -. < <. ) =.9544.343.343.477.477 -.. -...843 -.5.9544 =.477 +.477 P ( -3. < < 3. ) =.9974 P ( -.4 < <.33 ) =.993.4987.4987.49.49.88.99-3. 3. -.4.33 3 4 ND - 4
7//3 = = = 3 Stadard = 9 Dstrbuto Stadardze the - Stadardzed = = 5 = 7 5 = Oe table! 6 Theorem If s N(, ), the = ( ) s N(,). Stadardze the Dstrbuto N (, ) N (, ) Stadard Dstrbuto a - b - P ( a b) P b - a - F - F 7 a b a - a - b - P ( a b) P b - 8 Dstrbuto = Example For a ormal dstrbuto that has a mea = 5 ad s.d. =, what percetage of the dstrbuto s betwee 5 ad 6.? Dstrbuto = Example - 5-5 - 6. - 5. P(5 6.) P(.) Stadardzed = = 5 6. = 5 6. =. 9 3 ND - 5
7//3 Stadardzed Dstrbuto Table Example = Dstrbuto = Example - 5-5 - 6. - 5. P(5 6.) P(.).478 4.78% Stadardzed = =. Area =.5478 -.5 =.478 = 5 6. =. 3 3 Dstrbuto = Example P(3.8 5) - 3.8-5 -. P(3.8 5) =P(-. ).478 Stadardzed =.478 Dstrbuto = Example P(.9 7.) -.9-5 -. - 7.- 5. P(.9 7.) =P(-..).664 Stadardzed =.664 3.8 = 5 -. Area =.478 = 33.9 5 7. -.. 34 Area =.83 +.83 =.664 Example P( > 8) Example P( > 8) Dstrbuto = = 5-8 - 5.3 P( > 8) =P( >.3) =.38 8 Stadardzed = Area =.38 =.3.38 35 Dstrbuto = = 5 6% 8 38% Value 8 s the 6 d percetle 36 ND - 6
7//3 More o The work hours per week for resdets Oho has a ormal dstrbuto wth = 4 hours & = 9 hours. Fd the percetage of Oho resdets whose work hours are A. betwee 4 & 6 hours. P(4 6) =? B. less tha hours. P( ) =? Dstrbuto = 9 P(4 6) =? - 4-4 9-6- 4 9 P(4 6) P( ) =.477(47.7%).477 Stadardzed 37 4 6 4. 38 Dstrbuto = 9 P( ) =? - - 4 -.44 9 P( ) = P( -.44) =.73 =.73% Stadardzed = What s z gve P( < z) =.8?.8 Fdg Values for Kow Probabltes. Stadardzed Dstrbuto Table.73 4 -.44 4. 39 z =.84 z. =.84 Def. z a : P( z a ) = a ; P( < z a ) = a 4 Fdg Values for Kow Probabltes Example: The weght of ew bor fats s ormally dstrbuted wth a mea 7 lb ad stadard devato of. lb. Fd the 8th percetle. Area to the left of 8th percetle s.8. I the table, there s a area value.7995 (close to.8) correspodg to a z-score of.84. 8th percetle = 7 +.84 x. = 8.8 lb 4 Fdg Values for Kow Probabltes 7 Stadardzed.8.8 = 8.8 z =.84 (.84)(.) 8 th percetle 7 8.8 4 ND - 7
7//3 Stae Score Theorem 3 4 5 6 7 8 9? % 4% 4% If the radom varable s N(, ), >. the the radom varable ( ) V = = s c (). -.75 -.5 -.75 -.5.5.75.5.75 43 44 More Examples The pulse rates for a certa populato follow a ormal dstrbuto wth a mea of 7 per mute ad s.d. 5. What percet of ths dstrbuto that s betwee 6 to 8 per mute? The weghts of a populato follow a ormal dstrbuto wth a mea 3 ad s.d.. What percet of ths populato that s betwee ad 5 lbs? The Radom Fuctos Assocated wth s 45 46 Theorem (M.G.F.) Theorem (Lear Fucto) If,,, are depedet radom varables wth m.g.f. s (t), =,,,, M the the m.g.f. of Y a M ( t) Y M s ( a t) 47 If,,, are mutually depedet radom varables from ormal dstrbutos wth meas,, 3,,, ad varaces,, 3,,, the the lear fucto Y has the ormal dstrbuto N(Sc, Sc ). Y f c. c 48 ND - 8
7//3 Proof: M ( t) Y m.g.f. of e M ( c t) t c t c N e ct c t / c, c 49 Example: A elevator ca take lb maxmum, ad t s posted the elevator that the maxmum capacty s 5 adults. The weght dstrbuto of adults that use ths elevator s ormally dstrbuted wth mea =6 lb ad stadard devato = lb.. If a adult perso walk to ths elevator, how lkely that hs or her weght s over 5 lb?. If 5 adult persos walk to ths elevator, how lkely that the total weght wll be greater tha lb? (The same as the average s over. ) 5 Theorem (Dstrbuto of ) Theorem (Ch-square dstrbuto) If,,, are observatos of a radom sample of sze from the ormal dstrbuto N(, ), the the dstrbuto of the sample mea s N(, /) Stadard Error 5 If,,, have the stadard ormal dstrbuto N(, ). If these radom varables are depedet the W = + + + has a Ch-square dstrbuto wth degrees of freedom, c (). 5 Theorem (Dstrbuto of ad S ) Studet s t Dstrbuto If,,, are observatos of a radom sample of sze from the ormal dstrbuto N(, ). The statstcs, sample mea,, ad sample varace, S, are depedet ad ( -) S s c ( -) S ( - ) - 53 Let be a radom varable that s N(, ), ad U be a radom varable that s c (r), ad ad U are depedet. The, the radom varable T U r has a t-dstrbuto wth degrees of freedom r. 54 ND - 9
. 7 6 5 4 3 Std. Dev = 8.88 Mea =.3 N = 4. Std. Dev =.3 Mea = 9.84 N = 4. 7 6 5 4 3 Std. Dev = 5.4 Mea = 9.4 N = 4. Std. Dev =.64 Mea = 9.75 N = 4. 8 6 4 3 7//3 Std. Dev = 4.3 Mea = 9.9 N = 4. Std. Dev =. Mea = 9.8 N = 4. Studet s t Dstrbuto t-statstc If,,, are observatos of a radom sample of sze from the ormal dstrbuto N(, ). The statstcs, sample mea,, ad sample varace,, are depedet ad - T S has a t-dstrbuto wth d.f.. T - - ( -) S S ( -) d.f. of U U The 3 The Cetral Lmt Theorem 57 Theorem (Dstrbuto of ) If,,, are observatos of a radom sample of sze from a dstrbuto that has a mea ad a fte varace, the the dstrbuto of - s N(, ), as, / ad the dstrbuto of the sample mea s N(, /), as. - 58 A Radom Sample from Populato Smulated Samplg Dstrbuto of Meas 8 6 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. SIE = SIE4 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. =4 SIE 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. = 4 9. 8. 7. 6. 5. 4. 3... Std. Dev =.9.. Mea =.7 N = 4. Radom Sample of Sze 4 from Populato Populato mea = 9.9, stadard devato =.6 59 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. SIE5 =5 SIE5 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. =5 SIE 77. 73. 69. 65. 6. 57. 53. 49. 45. 4. 37. 33. 9. 5.. 7. 3. 9. 5.. = 6 ND -
7//3 Probablty Related to Mea Example: Cosder the dstrbuto of serum cholesterol levels for 4- to 7-year-old males lvg commuty A has a mea of mg/ ml, ad the stadard devato of 46 mg/ ml. If a radom sample of dvduals s take from ths populato, what s the probablty that the average serum cholesterol level of these dvduals s hgher tha 5? 6 P( > 5) =? N( x, x 4.6) = 5 P( > 5) P( > 3.4). 3.4 Cholesterol Level has a mea, s.d. 46.. x 5 - = 3.4 4.6 z 6 Approxmato for Bomal Probablty The 4 Approxmato for Dscrete Dstrbutos ( approxmato to Bomal) P( 3)? b = p = 3.5 = p(-p) =.75 3 4 5 6 7 63 64 Approxmato for Bomal Probablty P( b 3) P( N >.5).5-3.5 P( > ).75 P( > -.76).7764 = p = 3.5 = p(-p) =.75 Bomal Table: P( b 3) =.7734 3 4 5 6 7 65 ND -