chapter 1

Chapter 5: Joint Probability Distributions and Random Samples8.⎛8⎞⎛10⎞⎛12⎞⎝3⎠⎝2 ⎠⎝1 ⎠⎛30 ⎞⎜ ⎟ = 593,⎝ 6 ⎠a. numerator = ⎜ ⎟⎜⎟⎜⎟ = ( 56)( 45)( 12) = 30, 24030,240593,775denominator = 775 ; p(3,2) = = . 0509b. p(x,y) =⎧⎛8⎞⎛10⎞⎛12⎪⎜⎟⎜⎟⎜⎪⎝x⎠⎝y ⎠⎝6−⎨ ⎛30⎞⎪ ⎜ ⎟⎪ ⎝ 6 ⎠⎩ 0( x + y)⎞⎟⎠x,y _ are _ non − negativeint egers _ such _ that0 ≤ x + y ≤ 6otherwise9.= ∞ ∞30 3021 y ) dxdy−∞−∞20 202a.∫ ∫ f ( x,y)dxdy = ∫ ∫ K(x += K30303030210 10K20 ∫ y dy20222∫ ∫ x dydx + K∫ ∫ y dxdy = K∫x dx +203020⎛19,000⎞ 3= 20 K ⋅ ⎜ ⎟ ⇒ K =⎝ 3 ⎠ 380,00020302026262( x dx202 2b. P(X < 26 and Y < 26) =∫ ∫ K x + y ) dxdy = 12K∫c.2026204Kx2632030= 38,304K= .3024Iy = x+2y = x−2IIIII202030P( | X – Y | ≤ 2 ) =∫∫regionIII1f ( x,y)dxdy− ∫∫ ( x,y)dxdy −∫∫1−∫ ∫f f ( x,y)dxdyI28 3020 x+2IIf ( x,y)dydx −= (after much algebra) .359330 x−2∫ ∫2220f ( x,y)dydx178