DISTRIBUCIONES DE PROBABILIDAD CONJUNTA 1.Sea X el nmero de veces que falla cierta mquina de control numrico: 1, 2 o 3 veces en un da dado. Sea Y el nmero de veces que se llama a un tcnico para una emergencia. Su distribucin de probabilidad conjunta est dada como: p(x, y)x123 10.050.050.10 y20.050.100.35 300.200.10

a.Evale la distribucin marginal de X. b.Evale la distribucin marginal de Y. c.Encuentre P(Y = 3 / X = 2). Solucin: a. b.

c.P(Y = 3 /X =2) ( ) ( )( , ) (2, ), 2( ) (2)(2,1) 0.05 (2, 2) 0.10 (2,3) 0.200.1428, 0.2857, 0.5715(2) 0.35 (2) 0.35 (2) 0.35x xx x xpxy p yPYX PYXp x pp p pp p p= = == = = = = = P(Y = 3 /X = 2) = 0.5715 2.SealafuncindeprobabilidadconjuntadelasvariablesaleatoriasXyY,dada enlasiguiente tabla: x123 px (x) 0.100.350.55 y123 py (y)0.200.500.30y123 p (y/x=2) 0.14280.28570.5715 x y258 0.40.150.120.03 0.80.050.300.35

a.Calcule P(Y = 0.8 / X = 5) b.Calcule E (Y / X = 8) c.Calcule la covarianza y el coeficiente de correlacin. Solucin: a.

(5, ) (5, 0.4) 0.12 (5, 0.8) 0.30( / 5) , 0.2857, 0.7143(5) 0.42 0.42 0.42 0.42xp y p pP Y Xp= = = = = =

P(Y = 0.8 / X = 5) = 0.7143 b.(8, ) (8, 0.4) 0.03 (8, 0.8) 0.35( / 8) , 0.0789, 0.9211(8) 0.38 0.38 0.38 0.38xp y p pP Y Xp= = = = = = ( / 8) (0.4)(0.0789) (0.8)(0.9211) 0.7684 ( / 8) 0.7684 E Y X E Y X= = + = = = c.( , ), ( , ) ( ) ( ) ( )XYX YCovXYCovXY EXY EXE Y po o= = | | | |2 22 2( ) ( ) ( ) , ( ) ( ) ( ) , ( ), ( )X YVX EX EX VY E Y E Y VX VY o o = = = =

( ) 2(0.4)(0.15) 2(0.8)(0.05) 5(0.4)(0.12) 5(0.8)(0.30) 8(0.4)(0.03)8(0.8)(0.35) 3.976EXY = + + + ++ = ( ) 2(0.20) 5(0.42) 8(0.38) 5.54, ( ) 0.4(0.30) 0.8(0.70) 0.68 EX E Y = + + = = + = Cov.(X, Y) = (3.976) (5.54)(0.68) = 0.2088Cov.(X, Y) = 0.2088 222 2 22( ) 4(0.20) 25(0.42) 64(0.38) 35.62, ( ) 5.54,( ) 35.62 (5.54) 4.9284( ) (0.4) (0.30) (0.8) (0.70) 0.496, ( ) 0.68( ) 0.496 (0.68) 0.0336, 4.9284 2.22, 0.18330.20880.5131 (2.22)(0.1833)X YEX EXVXE Y E YVY o op= + + = == == + = == = = = == = 0.5131 p = x258 ( )xp x 0.200.420.38 y0.40.8 ( )yp y0.300.70 y0.40.8 ( / 5) py x =0.28570.7143 y0.40.8 ( / 8) py x =0.07890.9211 3.Sea X el tiempo de reaccin, en segundos, a cierto estimulante y Yla temperatura ( F ) a la que ciertareaccin comienzaasuceder.SupongaquelasdosvariablesaleatoriasXyYtienenla siguiente fdp conjunta: < < < = = = = = > = = 5.Sea X el dimetro de un cable elctrico blindado y Yel dimetro del molde cermico que tiene elcable.Xy Ytienenuna escala tal queestn entre 0 y 1. SupongaqueXy Y tienen unafdp conjunta:

< < ). b.Calcule la covarianza y el coeficiente de correlacin. Solucin: a. 121 1 14 2 21211 124 2121 14 2121 14 2112010111 1( )1 11 1 1( )2122y yyy yyPX Y dxdy dxdyy yx dy xy yy y dy y dyy ydy dyy+ > = += +| |= + + |\ .| |= + |\ .

1 12 21 1 14 4 211 1 12 4 2121 1 1 2 1 12 ln 2( ) ln (1 ) 0.3466 0.65342 2 1 4 2 2( ) 0.6534y y yPX Y| |= + = + = + = |\ .+ > = b.( , ), ( , ) ( ) ( ) ( )XYX YCovXYCovXY EXY EXE Y po o= = | | | |2 22 2( ) ( ) ( ) , ( ) ( ) ( ) , ( ), ( )X YVX EX EX VY E Y E Y VX VY o o = = = =

1 1 1 12 2 30 0 0 0 0 01 1 1 12 20 0 0 0 0 01 1 1 1200 0 0 0 01 1 1 1 1( )2 2 6 61 1 1 1 1( )2 2 4 41 1 1( )2 2y yy yyyEXY xy dxdy x dy ydy yyEX x dxdy x dy ydy yy yE Y y dxdy x dy ydy yy| |= = = = = |\ .| |= = = = = |\ .| |= = = = = |\ .

1 1 1 1.( , )6 4 2 24Cov XY| || |= = | |\ .\ .

1.( , )24Cov XY =

1 1 1 12 2 3 2 30 0 0 0 0 01 1 1 12 2 2 300 0 0 0 02 21 1 1 1 1( )3 3 9 91 1 1( )3 31 1 7 1 1 1( ) , ( ) , 0.22048, 0.288679 4 144 3 2 121 24(0.22048)(0.2886y yyyX YEX x dxdy x dy ydy yy yE Y y dxdy y x dy ydy yyVX VY o op| |= = = = = |\ .| |= = = = = |\ .| | | |= = = = = = ||\ . \ .= 0.6546 0.65467)p = =