Homework Set October 16, 2015
Solutions
Graded problems are in red
Chapter 4
Problem 40 Let X be the number of correct answers. Then
P cfw_X 4 = P cfw_X = 4 + P cfw_X = 5 =
1
11
5 1 2
.
+ 5 =
243
4 34 3 3
Problem 45 Let A be the event that the student ha
NAME
MATH 461  F13, Test 2, Fall 2015
November 13, 2015
Books, notes and extra papers are not allowed on this test
Please show all your work and explain all answers to qualify for full credit
1. (15 points) Consider a roulette wheel consisting of 38 numb
NAME
MATH 461  F13, Test 1, Fall 2015
October 9, 2015
Calculators, books, notes and extra papers are not allowed on this test
Please show all your work and explain all answers to qualify for full credit
1. (20 points) From a group of 7 freshmen, 6 sophom
Homework Set December 4, 2015
Solutions
Graded problems are in red
Chapter 7
Problem 5 If (X, Y ) is the location of the accident, then X and Y are uniform
3
random variables on ( 2 , 3 ). Let D = X + Y . Then
2
E cfw_D = E cfw_X + E cfw_Y  = 2E c
Homework Set November 20, 2015
Solutions
Graded problems are in red
Chapter 6
Problem 27 Let X1 , X2 be exponential random variables with parameter 1 , 2 . Let
X1
Z = X2 . Note that FZ (a) = 0 if a 0. Compute FZ (a) for a > 0:
FZ (a) = P cfw_Z a = P cfw_X
Homework Set November 6, 2015
Solutions
Graded problems are in red
Chapter 5
Problem 37 Let X be uniformly distributed over (1, 1).
(a) P X >
1
2
=P X>
1
2
+ P X < 1 =
2
1
2
(b) Let Y = X. If y (0, 1), then FY (y) = P cfw_Y y = P cfw_y Y y =
y, so tha
Homework Set October 23, 2015
Solutions
Graded problems are in red
Chapter 4
Problem 84 (a) For i = 1, . . . , 5, let Xi = 1 if the ith box is empty and Xi = 0
otherwise. Then X = X1 + + X5 is the number of empty boxes. For
i = 1, . . . , 5,
E[Xi ] = P (
9th Homework Set Solutions
Chapter 6
Problem 6.11 Let A be the number of people buying an ordinary set, B the number
of people buying a plasma set, and C the number of people who are
5!
just browsing. Then P cfw_A = 2, B = 1, C = 2 = 2!1!2! 0.452 0.150.42
Seventh Homework Set Solutions
Chapter 5
Problem 5.6 (a)
E [X] =
xf (x)dx =
1
4
x
x2 e 2 dx
0
x
1
2x2 8x 16 e 2  = 4
=
0
4
(b) E [X] =
(c) E [X] =
1
1
5
c(1 x2 )xdx = 0 by symmetry
5
x x2 dx =
5
5 x
=
Problem 5.10 (a) Let X be uniform on [0, 60]. Then
P
10th Homework Set Solutions
Chapter 6
Problem 6.48 Let X1 , . . . , X5 be independent exponential random variables with parameter .
(a)
P cfw_min(X1 , . . . , X5 ) a = 1 P cfw_min(X1 , . . . , X5 ) > a
= 1 P cfw_X1 > a, . . . , X5 > a
= 1 P cfw_X1 > a P c
Homework Set January 29, 2016
Solutions
Chapter 1
Graded problems are in red
Problem 2 Each roll of one die has 6 possible outcomes. Since there are 4 rolls,
there are 64 = 1, 296 possible outcome sequences.
Problem 4 If each of the boys can play any inst
Math 461, Section P1
Homework 6
Will not be collected.
Midterm I: Make sure you know how to correctly and completely solve any
problem from midterm I, their statements can be found on the course
web page;
Section 4.1 textbook pages 172173 Exercises: 4.1,
Review Final
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11th Homework Set Solutions
Chapter 7
Problem 7.50 We have
x
e y y
dx = ey
y
fY (y) =
0
for y > 0, so that
e
x>0
0 x 0.
fXY (xy) =
y
Now, we have
E X 2 Y =
0
Problem 7.51 We have
y
fY (y) =
0
x
y
x2 x
e y dx = 2y 2 .
y
ey
dx = ey ,
y
so that
1
y
x (0,
12th Homework Set Solutions
Chapter 8
Problem 8.1 P (0 X 40) = 1 P (X 20 > 20) 1
Problem 8.2 (a) P (X 85)
75
85
20
400
=
19
.
20
15
.
17
=
(b) P (65 X 85) = 1 P (X 75 > 10) 1
25
100
= 3.
4
(c) Since
n
P

i=1
Xi
75 > 5
n
25
,
25n
we need n = 10.
Lesson 1: Tools for data analysis
Tutorials
T.1) Basic measurements you can make with cumdist[x, X] and Freq[x, X]
T.1.a)
Here's a data set X:
X = Sort@842., 45., 73., 78., 54., 22., 47., 83., 33., 47., 67., 43., 47., 63., 55., 60., 54., 55.,
53., 60., 5
Lesson 1: Tools for data analysis
Try It
The cell below will be filled in by the grader:
Correctness/Completeness: 10/10
Quality of Explanations: 10/10
Total Score: 20/20
Mentor Response
Good job, no point deducted
G.7) Dice
G.7.a.i) The dicedata set
Whe
Lesson 2: Probabilities
Tutorials
T.1) Probability calculations in context: Demographics
This problem was adapted from Jim Pitman's book,
"Probability" (Springer Texts in Statistics, SpringerVerlag New York, 1993).
T.1.a.i) Rich or famous or both?
A soc
Lesson 2: Probabilities
Basics
B.1) Prob[A, X]: The probability that a pull from X is in a set A
B.1.a.i) A probabilitycalculation
Here is a little data set X:
X = Table@Random@Integer, 80, 5<D, 8j, 1, 27<D;
Xfreqplot = Show@FreqPlot@ X, Purple, Darker@Y
Lesson 1: Tools for data analysis
B.1) Data analysis: Frequency plots
B.1.a.i) Frequency plots for data sets
Here's a simple data set X:
X = 82., 3., 3., 4., 4., 5., 5., 5., 5., 6.<
82., 3., 3., 4., 4., 5., 5., 5., 5., 6.<
Here's what lots of folks call