chapter 1

CHAPTER 5Section 5.11.a. P(X = 1, Y = 1) = p(1,1) = .20b. P(X ≤ 1 and Y ≤ 1) = p(0,0) + p(0,1) + p(1,0) + p(1,1) = .42c. At least one hose is in use at both islands. P(X ≠ 0 and Y ≠ 0) = p(1,1) + p(1,2) + p(2,1)+ p(2,2) = .70d. By summing row probabilities, p x (x) = .16, .34, .50 for x = 0, 1, 2, and by summingcolumn probabilities, p y (y) = .24, .38, .38 for y = 0, 1, 2. P(X ≤ 1) = p x (0) + p x (1) = .50e. P(0,0) = .10, but p x (0) ⋅ p y (0) = (.16)(.24) = .0384 ≠ .10, so X and Y are not independent.2.a.p(x,y) 0 1 2 3 40 .30 .05 .025 .025 .10 .5x 1 .18 .03 .015 .015 .06 .3y2 .12 .02 .01 .01 .04 .2.6 .1 .05 .05 .2b. P(X ≤ 1 and Y ≤ 1) = p(0,0) + p(0,1) + p(1,0) + p(1,1) = .56= (.8)(.7) = P(X ≤ 1) ⋅ P(Y ≤ 1)c. P( X + Y = 0) = P(X = 0 and Y = 0) = p(0,0) = .30d. P(X + Y ≤ 1) = p(0,0) + p(0,1) + p(1,0) = .533.a. p(1,1) = .15, the entry in the 1 st row and 1 st column of the joint probability table.b. P( X 1 = X 2 ) = p(0,0) + p(1,1) + p(2,2) + p(3,3) = .08+.15+.10+.07 = .40c. A = { (x 1 , x 2 ): x 1 ≥ 2 + x 2 } ∪ { (x 1 , x 2 ): x 2 ≥ 2 + x 1 }P(A) = p(2,0) + p(3,0) + p(4,0) + p(3,1) + p(4,1) + p(4,2) + p(0,2) + p(0,3) + p(1,3) =.22d. P( exactly 4) = p(1,3) + p(2,2) + p(3,1) + p(4,0) = .17P(at least 4) = P(exactly 4) + p(4,1) + p(4,2) + p(4,3) + p(3,2) + p(3,3) + p(2,3)=.46175