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Rice - Mathematical Statistics and Data Analysis 3 (Homework)

James Finch

Statistics, section 2, Fall 2019

Instructor: Dr. Friendly

Current Score : 3 / 34

Due : Sunday, January 27, 2030 23:30 EST

Last Saved : n/a Saving...  ()

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Question
Points
1 2 3 4 5 6
–/2 3/12 –/5 –/3 0/10 –/2
Total
3/34 (8.8%)

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. –/2 points RiceMathStat3 1.P.047. My Notes
Question Part
Points
Submissions Used
1 2
–/1 –/1
0/50 0/50
Total
–/2
 
Urn A has five red, five blue, and two green balls. Urn B has five red, two blue, and four green balls. A ball is drawn from urn A and put into urn B, and then a ball is drawn from urn B.
(a) What is the probability that a red ball is drawn from urn B?


(b) If a red ball is drawn from urn B, what is the probability that a red ball was drawn from urn A?
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2. 3/12 points  |  Previous Answers RiceMathStat3 1.P.065. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10 11 12
0/1 0/1 1/1 0/1 1/1 0/1 0/1 0/1 0/1 0/1 1/1 0/1
1/50 1/50 1/50 1/50 1/50 1/50 1/50 1/50 1/50 1/50 1/50 1/50
Total
3/12
 
Show that if A and B are independent, then A and Bc as well as Ac and Bc are independent.

A and B are independent right double arrow implies P(A ∩ B) = P(A)P(B)
P(A ∩ B) = P(A)P(B)
= P(A)( Incorrect: Your answer is incorrect. )
= P(A) − P(A) Incorrect: Your answer is incorrect.
P(A) Correct: Your answer is correct. = P(A) − P(A ∩ B)
= P(A ∩ B) + Incorrect: Your answer is incorrect. − P(A ∩ B)
= Correct: Your answer is correct.
right double arrow implies A and Bc are independent.

P(A ∩ B) = P(A)P(B)
= ( Incorrect: Your answer is incorrect. )(1 − P(Bc))
= 1 − P(Ac) − P(Bc) + Incorrect: Your answer is incorrect.
Incorrect: Your answer is incorrect. = P(Ac) + P(Bc) − 1 + P(A ∩ B)
= Incorrect: Your answer is incorrect. + 1 − P(B) − 1 + P(A ∩ B)
= 1 − ( Incorrect: Your answer is incorrect. + P(B) − P(A ∩ B))
= 1 − Correct: Your answer is correct.
= Incorrect: Your answer is incorrect.
right double arrow implies Ac and Bc are independent.
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3. –/5 points RiceMathStat3 2.P.034. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5
–/1 –/1 –/1 –/1 –/1
0/50 0/50 0/50 0/50 0/50
Total
–/5
 
Let
f(x) = 
(1 + αx)
2
 for −1 ≤ x ≤ 1 and f(x) = 0 otherwise, where −1 ≤ α ≤ 1.

Show that f is a density.

This answer has not been graded yet.



Find the corresponding cdf.
F(x) =


Find the quartiles and the median of the distribution in terms of α.
x1/4 =
x1/2 =
x3/4 =
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4. –/3 points RiceMathStat3 2.P.053. My Notes
Question Part
Points
Submissions Used
1 2 3
–/1 –/1 –/1
0/50 0/50 0/50
Total
–/3
 
Let X be a normal random variable with μ = 3 and σ = 10.
(a) Find P(X > 6).


(b) Find P(-12 < X < 9).


(c) Find the value of x such that P(X > x) = 0.05.
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5. 0/10 points  |  Previous Answers RiceMathStat3 3.P.017. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10
0/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1
2/50 0/50 0/50 0/50 0/50 0/50 0/50 0/50 0/50 0/50
Total
0/10
 
Let (X, Y) be a random point chosen uniformly on the region R = {(x, y) : |x| + |y| ≤ 1}.
(a) Sketch R.

Submission Data

Incorrect: Your answer is incorrect.
(b) Find the marginal densities of X and Y using your sketch. Be careful of the range of integration.
fX(x) =

,   ≤ x ≤
fY(y) =

,   ≤ y ≤

(c) Find the conditional density of Y given X.
fY|X(y|x) =
,
≤ y ≤
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6. –/2 points RiceMathStat3 3.P.066. My Notes
Question Part
Points
Submissions Used
1 2
–/1 –/1
0/50 0/50
Total
–/2
 
Each component of the following system (see figure below) has an independent exponentially distributed lifetime with parameter λ. Find the cdf and the density of the system's lifetime. (Note:
E(X) = 
1
λ
.
)
F(t) =
f(t) =
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