Download ECE316 Tutorial 9

ECE316 Tutorial 9 Chapter 6, Problems 6.2, 6.3, 6.4, 6.8 and 6.9 Review ,
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Problem 6.2 – Page 287 Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let Xi equal 1 if the ith ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of: (a) X1 , X2 (b) X1 , X2 , X3 Solution: (a)
0,
0
0,
1
1,
0
1,
1
(b) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Joint Probability 8.7.6
13.12.11
8.7.5
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8.7.5
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8.5.4
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8.7.5
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8.5.4
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8.5.4
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5.4.3
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Problem 6.3 – Page 287 In problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Find the joint probability mass function of (a) Y1 , Y2 (b) Y1 , Y2 , Y3 Solution: (a) 0 0 Joint Probability 11.10.9
13.12.11
0 1 1
13
11.10
12.11
3 1 0 1
13
11.10
12.11
3 1 1 1
13
1
12
11
11
3
2 (b) 0 0 0 0 0 1 1.10.9
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3 0 1 0 1.10.9
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3 0 1 1 1 0 0 1 0 1 1.1.10
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3
2 1 1 0 1.1.10
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3
2 1 1 1 Joint Probability 10.9.8
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1.1.10
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3
1.10.9
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1.1.1
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3
2 3 2
1 Problem 6.4 – Page 287 Repeat problem 2 when the ball selected is replaced in the urn before the next selection. Solution: (a)
0,
0
0,
1
1,
0
1,
1
(b) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Joint Probability 8
13
8
5
13
13
8
5
13
13
5
8
13 13
8
5
13
13
5
8
13 13
5
8
13 13
5
13
Problem 6.8 – Page 287 The joint probability density function of X and Y is given by ,
,0
∞ (a) Find c. (b) Find the marginal densities of X and Y. (c) E[X] Solution: .
(a)
.
.2
.
0
1 Ö
Using integration by parts... Let Then 3
.
4
3
4
3
4c
|
3
e
y
dy Using integration by parts... Let u y and dv e . dy Then du 2ydy and v
e 4c
|
Using integration by parts... Let u y and dv
Then du dy and v
8c
2
y
e
e
|
e
e
dy
0
∞ y
e
dy 8c
1 . dy dy
8c
1
8
(b)
8c
0
1
∞ 10
∞
9
8
7
6
5
4
3
2
1
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-∞
∞
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Problem 6.9 – Page 287 The joint probability density function of X and Y is given by ,
(a)
(b)
(c)
(d)
(e)
(f)
6
7
0
2
1 , 0
2 Verify that this is indeed a joint density function. Compute the density function of X. Find P{X>Y} Find P{Y > ½ | X< ½ } Find E[X] Find E[Y] Solution: (a) To be a probability density function, it has to be positive for all possible values of the 2 random variables, and the Integration over the whole range of X and Y should be “1”. 0
2
2
(b)
2 4
0
1 (c)
Or we can do it the other way. (d)
1 ,
|
,
2
,
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(e)
.
(f)
.
2