Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght, stress, cost, etc. The put varables to the model clude both desg varables ad desg parameters. Desg varables are those that are cotrollable by egeers, such as materal types ad dmesos. Desg parameters are ot cotrollable, for example, evrometal temperature, usage codtos, etc. If the put varables are radom, the respose varable wll also be radom. Aalyss Model =g() Fgure 5. Egeerg Aalyss Model It s crucal to evaluate the probablstc characterstcs of respose varables gve those of put varables. Ths helps egeers uderstad the mpact of the ucertaty assocated wth the put varables o the respose varables. For example, the maxmum stress that a compoet s subject to s oe of respose varables, ad the appled exteral force s oe of put varables. Egeers are terested kowg the dstrbuto of the maxmum stress whe they have formato about the dstrbuto of the exteral force. Wth the kowledge about how the maxmum stress vares due to the radomess of the exteral force, egeers wll be able to make approprate decsos to accommodate the varatos the maxmum stress. Mathematcally, the task s to evaluate the dstrbuto of a respose varable gve the dstrbutos of put varables. I ths chapter, we wll dscuss the theoretc dervato of probablty dstrbutos ad statstcal momets of a respose varable from the dstrbutos of put varables. Although the methodologes preseted ths chapter may ot be drectly applcable ad practcal to real-world applcatos, the dscusso wll serve as a theoretc foudato to egeerg ucertaty aalyss ad desg uder ucertaty. 5. Fuctos of a Sgle Radom Varable We wll frst dscuss a fucto of oly oe radom varable ad the exted the dscusso to geeral stuatos multple radom varables are volved.
Probablstc Egeerg Desg 5.. Lear Relatoshp Assume that radom varable s a lear fucto of radom varable ad the fuctoal relatoshp s gve by a ad b are costats. = a+ b (5.) Sce has a lear relatoshp wth, has the same dstrbuto as, but dfferet dstrbuto parameters, such as mea ad varace. The cdf of s ya ya F( y) = P ( y) = Pa ( + b y) = P ( ) = F ( ) (5.) b b The above equato shows that the cdf of has the same fuctoal form as. The pdf of ca also be wrtte terms of the pdf of as df ( y) y a f( y) = = f( ) (5.3) dy b b Based o Eq. 5.3, the mea of ca be derved from that of as = a+ b (5.4) ad the stadard devato of ca be derved from that of as = b (5.5) If follows a ormal dstrbuto,.e. N (, ), wll also follow a ormal dstrbuto ad N(, ) = Na ( + b, b ). Example 5. For example, the tolerace of the legth of a rectagular plate s large s assumed to be ormally dstrbuted wth N(0,0.5) cm. Sce the tolerace of the wdth s small, t s treated as a determstc quatty wthout ay radomess. The wdth s equal to 4 cm. The permeter of the plate s = + 8. Determe the dstrbuto of. The mea value of s = a+ b = 8+ 0= 8 cm
Fuctos of Radom Varables The stadard devato of s = b = 0.5= cm Sce s ormally dstrbuted, s also ormally dstrbuted. Its dstrbuted s N(8, ) cm. 5... Nolear Relatoshp If radom varable s a olear fucto of radom varable ad = g( ), the cdf of s gve by The pdf of s gve by [ ] g( x) F ( y ) = P ( y ) = P g ( ) y = f ( xdx ) (5.6) d d f( y) = [ F( y) ] = f( xdx ) dy dy g( x) y (5.7) Eqs. 5.6 ad 5.7 are applcable to ay cotuous fucto = g( ). If radom varable s a mootocally creasg or decreasg fucto of radom varable, Eqs. 5.6 ad 5.7 ca be evaluated coveetly. As show Fg. 5., sce s a mootocally creasg or decreasg terms of, = g ( ) wll be a sglevalued fucto of, ad y s equvalet to x. Therefore, y F( y) = P ( y) = P ( x) = P g ( y) = F[ g ( y)] (5.8) y y y y x (a) x = g ( y) x x = g ( y) (b) Fgure 5. (a) Mootocally Icreasg ad (b) Decreasg Fuctos 3
Probablstc Egeerg Desg The pdf of s derved as df( y) df( x) dx dx dg ( y) ( ) ( ) [ ( )] f y = = = f x = f g y (5.9) dy dx dy dy dy dg ( y) Sce a pdf s oegatve ad the dervatve dy dg ( y) of s used. Eq. 5.9 s the rewrtte as dy ca be egatve, the absolute value dg ( y) f( y) = f[ g ( y)] (5.0) dy Example 5. If the dameter of the crcular cross secto of a trasmsso shaft s = D ~ N (, ) (see Fg. 5.3), what s the probablty desty fucto of the cross sectoal area = A = g ( ) =? 4 C A B A - A D A Fgure 5.3 A Trasmsso Shaft The fucto g A 4 = ( ) = = s show Fg. 5.3. 4
Fuctos of Radom Varables y y x = y Fg. 5.3 Fucto = g( ) = 4 Fg. 5.3 graphcally suggests that y s equvalet to y, ad therefore, F( y) = P ( y) = P y = F y Dfferetatg the cdf gves the pdf f( y) = f y y Sce f x ( x) = exp the pdf of s the gve by y f ( y) exp = y If the dstrbuto of the dameter s N(50, ) mm, the above equato gves the followg dstrbuto of the cross sectoal area. 5
Probablstc Egeerg Desg f ( y) = exp y 50 y The pdfs of ad are depcted Fg. 5.4. The same result ca be obtaed by usg Eq. 0 drectly. g ( y) = y dg ( y) = dy y Usg Eq. 0 yeld the same pdf f( y) = f y y (a) (b) Fgure 5.4 (a) pdf of ad (b) pdf of 5.3 Fuctos of Several Radom Varables Cosder a fucto of radom varables (,,, ) g = (,,, ) (5.) 6
Fuctos of Radom Varables If the jot pdf of (,,, ) s f,,, ( x, x,, x ), the pdf of the fucto s gve by F ( y) P ( y) f ( x, x,, x ) dx dx, dx (5.) = =,,, g( x, x,, x ) y For a geeral olear fucto egeerg applcatos, t s very dffcult or eve mpossble to use the above equato to obta the cdf of the respose varable. I ucertaty aalyss that wll be preseted later ths book, we wll dscuss approxmato methods to the probablty tegrato Eq. 5.. It s possble to use Eq. 5. for some specal cases. For example, f s a lear combato of depedet ormal varables ~ N(, ), =,,,, the 0 = a + a (5.3) = whch a are costats, t ca be show that s also ormally dstrbuted wth the followg mea ad varace ad 0 = = a + a (5.4) a = = (5.5) Example 5.3 As show Fg. 5.5, three torques that exert to a trasmsso shaft are ormally dstrbuted wth = T ~ N(, ) (500,0) N m = N, = T ~ N (, ) = N(50,5) N m, ad ~ T = N(, ) = N(300,30) N m. What s the dstrbuto of the resultat torque? 3 3 3 3 7
Probablstc Egeerg Desg T 3 T Fgure 5.5 A Trasmsso Shaft The total torque T s the sum of the three dvdual torques,.e. = T = T + T T3 = + 3 s also ormally dstrbuted wth the followg mea ad stadard devato. T = + + = 500+ 50 300= 350 N m 3 Ad = + + = 0 + 5 + 30 = 36.40 N m 3 5.4 Momets of a Fucto of Several Radom Varables As see Secto 5.3, t s dffcult to obta the cdf or pdf of a respose varable whch s a geeral fucto of radom varables. However, t s relatvely easy to obta the momets of the respose varable for some specal fuctos. 5.4. Mea ad Varace of a Lear Fucto If s a lear fucto of = (,,, ) wth the followg equato 0 = a + a (5.6) whch a are costats, smlar to the dervato of Eqs. 5. ad 5.3, the mea ad varace of are gve by = 0 = = a + a (5.7) 8
Fuctos of Radom Varables s the mea of, ad = a + aa j j j = = j= j (5.8) ρ s the varace of ad ρ j s the correlato coeffcet betwee ad j. If (,,, ) are mutually depedet, Eq. 5.8 becomes a = = (5.9) 5.4. Other Commo Equatos The momets of several commo fuctos are provded below. γ deotes the coeffcet of skewess. δ deotes the coeffcet of varato ad s gve by δ = (5.0) ) = a + b + c 4 = ( a + b)[ a( + γ) + b] + a (4+ 3 γ ) (5.) 3 3 γ = ( ( a + b)[( a( + b) γ + a (4+ 3 γ ) (5.) ) = a 3 4 = a [ + ( ) δ + ( )( ) δ γ + ( )( )( 3)( + γ ) δ] 6 6 (5.3) = ( aδ ) A (5.4) γ a B = sg. A 3/ (5.5) A= + ( ) δγ + ( )(3 5) δ + ( )(7 ) δγ (5.6) 8 9
Probablstc Egeerg Desg 3) a = + b 3 ( )(4 3 ) B= γ + + γ δ (5.7) 4 a 3 = + δ δ γ + δ + δ γ + b (5.8) 3 4 4 ( 3 ) x a = A (5.9) δ B γ = sg( a) (5.30) A 3/ 9 A = δ γ + 8δ + δγ (5.3) 4) Z = a + b 9 B = 6δ γ + δγ (5.3) 3 = + δ δγ + δ + δγ + b (5.33) 3 4 4 Z a ( 3 ) Z = a A (5.34) B γ Z = sg( a) (5.35) A 3/ 3 4 9 4 A = δ + δ δγ + 8δ + 3δ δ + δγ (5.36) 3 3 4 9 4 B = δγ δγ + 6δ + 6δδ + δγ (5.37) 5) = a + b = = a + b (5.38) = 0
Fuctos of Radom Varables a = = (5.39) 3 3 3 a = γ = γ (5.40) 6) = a + b = b (5.4) = = a γ + = a = A (5.4) B γ = sg( a) (5.43) A 3/ j = = j=+ (5.44) A = δ + δ δ 3 6 j = = j=+ (5.45) B = δ γ + δ δ 5.5 Cocludg Remarks Quatfyg the ucertaty of respose varables gve the ucertaty of put varables s oe of the most mportat tasks may egeerg desg applcatos, such as relablty-based desg, robust desg, ad desg for Sx Sgma. Ths ca help egeers uderstad the mpact of ucertaty assocated wth put varables o respose varables. Quatfyg the ucertaty of respose varables therefore ads egeers to make proper decsos to mtgate the effects of put ucertaty. Ths chapter provdes a fudametal troducto about how to evaluate the radomess of respose varables from the dstrbutos of put varables. The methods dscussed ths chapter serve as a theoretc foudato for ucertaty aalyss although they may ot be drectly applcable to real egeerg problems. I egeerg applcatos, respose varables are usually olear fuctos ad volve a large umber of radom put varables. More practcal methods for egeerg applcatos wll be dscussed later the followg chapters.