chapter 1

Chapter 5: Joint Probability Distributions and Random Samples19.a.f2 2f ( x,y)k(x + y )( y | x)= =2f ( x)10kx+ .05Y | X20 ≤ y ≤ 30X2 2k(x + y )⎛ 3 ⎞X |( x | y)=20 ≤ x ≤ 30 ⎜k= ⎟210ky+ .05⎝ 380,000 ⎠fYb. P( Y ≥ 25 | X = 22 ) =∫ 3030f Y25∫fY| Xy | 22)25( dy2 230 k((22)+ y )=∫ dy = .78325 210k(22)+ .052P( Y ≥ 25 ) = ( y)dy (10ky+ .05) dy = . 75∫ ∞ −∞= ∫30252k((22)2+ y )10k(22)+ .0530Y|X( ∫202c. E( Y | X=22 ) = y ⋅ f y | 22) dy = y ⋅dy= 25.3729122 2E( Y 2 302 k((22)+ y )| X=22 ) =∫ y ⋅dy = 652. 02864020210k(22)+ .05V(Y| X = 22 ) = E( Y 2 | X=22 ) – [E( Y | X=22 )] 2 = 8.24397620.f ( x1, x2,x3)x 3 | x1,x23 1 2 =f ( x , xwhere f,(1,2) =1 xx x2)a. f ( x | x , x )x1,x 212xthe marginal joint pdf∫ ∞ −∞f ( x dx1, x , x )of (X 1 , X 2 ) =2 3 3b. f ( x , x | x )f ( x , x, x1 2 3x 2 , x3|x12 3 1 = wherefx( x1)1f x∞ ∞1) = ∫−∞∫f ( x1, x2,x3)dx23−∞( xdx1)21. For every x and y, f Y|X (y|x) = f y (y), since then f(x,y) = f Y|X (y|x) ⋅ f X (x) = f Y (y) ⋅ f X (x), asrequired.183