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**Additional info for Instructor's Solutions Manual for Probability and Statistical Inference (8th ed.)**

**Example text**

2 2 = 1 (b) µX = 1 2 dx = 1 2 dy = x x+ 0 1 (c) µY = y y+ 0 1 (d) E(X 2 ) x2 x + = 0 2 σX = 1 3 1 2 x + x 3 4 1 = 0 7 ; 12 7 ; 12 1 2 E(X 2 ) − µ2X = dx = 5 − 12 1 4 1 3 x + x 4 6 7 12 2 = 1 = 0 5 , 12 11 . 144 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 59 Distributions of Two Random Variables (e) Similarly, σY2 = 11 . 2 1 dt1 dt2 20 (4 − t2 )2 40 4 = 2 1 . 8x. 2–4 E[a1 u1 (X1 , X2 ) + a2 u2 (X1 , X2 )] = [a1 u1 (x1 , x2 ) + a2 u2 (x1 , x2 )]f (x1 , x2 ) (x1 ,x2 ) ∈R = (x1 , x2 ) ∈R = u2 (x1 , x2 )f (x1 , x2 ) u1 (x1 , x2 )f (x1 , x2 ) + a2 a1 (x1 , x2 ) ∈R a1 E[u1 (X1 , X2 )] + a2 E[u2 (X1 , X2 )].

00405. 4! 7! (7 − x − y)! 1–10 (a) x = 0, 1, . . , 7. 1 2 1 2 P 0≤X≤ x2 0 1 2 0 1 (b) P 1 = = 1 2 ≤Y ≤1 √ y 1 2 0 1 1 2 3 dy dx 2 11 3 (1 − x2 ) dx = ; 2 16 = = 3 dx dy 2 3√ y dy = 1 − 2 1 2 P 1 2 ≤ X ≤ 1, ≤Y ≤1 = 1 2 1 = 1 2 = (d) P (X ≥ 12 , Y ≥ 21 ) 1 2 1 2 = P ( 21 ≤ X ≤ 1, = 5 − 8 1 2 1 2 ; 3 dx dy 2 √ 3 2 5 − 8 3/2 1 2 √ y 1 (c) 0 ≤ x + y ≤ 7; 1 2 y− dy 3/2 ; ≤ Y ≤ 1) 3/2 . (e) X and Y are dependent. 1–12 (a) f1 (x) = (x + y) dy 0 1 xy + y 2 2 1 1 0 ≤ x ≤ 10; =x+ , 2 0 1 1 f2 (y) = (x + y) dx = y + , 0 ≤ y ≤ 1; 2 0 1 1 f (x, y) = x + y = x + y+ = f1 (x)f2 (y).

2 e−λ [(4/3)λ − 1] λ3 e−λ 3! = 4· = 0 λ = 3/4 σ2 3 4 = E(X 2 ) − µ2 2 3 E(X 2 ) − 4 12 21 9 + = . 205. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 5116. 1–4: Weights of laptop computers x 27 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 2. The answer depends on the cost of the nails as well as the time and distance required if too few nails are purchased. 1–8: Melting points of metal alloys There are 31 observations within one standard deviation of the mean (62%) and 48 observations within two standard deviations of the mean (96%).