# Functions of Random Variables

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2 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

4 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 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

6 Probablstc Egeerg Desg f ( y) = exp y 50 y The pdfs of ad are depcted Fg 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

7 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?

8 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 = + + = = 350 N m 3 Ad = + + = = N m 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 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

9 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

10 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 ) 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) Z a ( 3 ) Z = a A (5.34) B γ Z = sg( a) (5.35) A 3/ A = δ + δ δγ + 8δ + 3δ δ + δγ (5.36) B = δγ δγ + 6δ + 6δδ + δγ (5.37) 5) = a + b = = a + b (5.38) = 0

11 Fuctos of Radom Varables a = = (5.39) 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.

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