175
CHAPTER 5
Section 5.1
1.
a.
P(X = 1, Y = 1) = p(1,1) = .20
b.
P(X
≤
1 and Y
≤
1) = p(0,0) + p(0,1) + p(1,0) + p(1,1) = .42
c.
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) = .70
d.
By summing row probabilities, p
x
(x) = .16, .34, .50 for x = 0, 1, 2, and by summing
column probabilities, p
y
(y) = .24, .38, .38 for y = 0, 1, 2.
P(X
≤
1) = p
x
(0) + p
x
(1) = .50
e.
P(0,0) = .10, but p
x
(0)
⋅
p
y
(0) = (.16)(.24) = .0384
≠
.10, so X and Y are not independent.
2.
a.
y
p(x,y)
0
1
2
3
4
0
.30
.05
.025
.025
.10
.5
x
1
.18
.03
.015
.015
.06
.3
2
.12
.02
.01
.01
.04
.2
.6
.1
.05
.05
.2
b.
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) = .30
d.
P(X + Y
≤
1) = p(0,0) + p(0,1) + p(1,0) = .53
3.
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 = .40
c.
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) =.22
d.
P( exactly 4) = p(1,3) + p(2,2) + p(3,1) + p(4,0) = .17
P(at least 4) = P(exactly 4) + p(4,1) + p(4,2) + p(4,3) + p(3,2) + p(3,3) + p(2,3)=.46