Chapter 2
Probability
is the measure of ones belief that a future (random)
:
event will occur.
A random
event is one whose occurrence cannot be predicted
:
with certainty.
However, we can often understand the long-run :
relative
frequency (proportion)
Math 447 - September 8, 2011 - Quiz 5 Solutions
Name:
1. The proportions of blood phenotypes A, B, AB, and O in the population of all Caucasians
in the United States are approximately 0.41, 0.10, 0.04, and 0.45 respectively. A single
Caucasian is chosen a
Math 447 - September 2, 2011 - Quiz
Name:
Express your answers to the following questions as numbers, not binomial coefficients or
sums thereof.
1. How many ordered triples of non-negative integers (a, b, c) satisfy a + b + c = 9?
9+31
11
=
= 55
31
2
2.
Math 447 - August 31, 2011 - Quiz
Name:
Let S = cfw_1, 2, 3, 4, 5, 6, 7. Express your answers to the following questions as numbers, not
binomial coefficients or sums thereof.
1. How many subsets of S have exactly 4 elements and contain cfw_1, 2?
10
2. Ho
Math 447 - September 1, 2011 - Quiz
Name:
Let S = cfw_1, 2, 3, 4, 5, 6, 7, 8. Express your answers to the following questions as numbers,
not binomial coefficients or sums thereof.
1. How many subsets of S have exactly 4 elements and contain cfw_1, 2?
15
Math 447 - September 7, 2011 - Quiz Solutions
Name:
Express your answers to the following questions as numbers, not binomial coefficients or
sums thereof.
1. How many distinct sequences of letters can be formed by rearranging the letters in the
word STATI
Class Work Problem Set 1
1. A clever machinist wants to mess with a young engineer. The young
engineer has designed a great gadget, which is supposed to detect
defective ball bearings. The machinist has an urn of 10 ball bearings in
which there are 3 defe
Math 447
Final Exam
Fall 2015
No books, no notes, only SOA-approved calculators. Please put your answers in the spaces provided!
Section:
Name:
Question Points Score
1
8
2
6
3
10
4
19
5
9
6
10
7
14
8
14
9
23
10
14
Total:
127
1. (8 points) (Monty Hall, Aga
Math 447
Test 2 Solutions
1. Let the random variable Y have distribution function
0,
y
,
8
F (y) = y2
16 ,
1,
Fall 2015
y 0,
0 < y < 2,
2 y < 4,
y 4.
(a) (5 points) What is the probability density function of
in the denition of F !)
Solution. This is ex
Homework 5 Solution
3.167
a. See the lecture notes.
b. Note that it is given that = 11 and 2 = 9. Using Tchebysheff(Chebysheff)s inequality, we
have that
V ar(Y )
9
P (|Y 11| C) = P (|Y EY | C)
= 2.
C2
C
To make sure that P (|Y 11| > C) 0.09, we just nee
Homework 4 Solution
3.127 Let Y be the number of typing errors in a page. Then Y Poisson() with = 4.
4
X
4y 4
103 4
P (Y 4) = (
)e = (1 + 4/1 + 16/2 + 64/6 + 256/24)e4 =
e = 0.6288.
y!
3
y=0
3.136 (a). Let X=number of cases of E.coli, then X is Poisson(1.
Homework 3 Solution
3.4 Define events Ak =cfw_the kth valve correctly opens, k = 1, 2, 3. Then
c
c
c
c
c
c
c
c
c
c
c
P (Y = 0) = P (A1 (A2 A3 ) = P (A1 )P (A2 A3 ) = P (A1 )[P (A2 ) + P (A3 ) P (A2 A3 )] = 0.072;
c
c
c
P (Y = 1) = P (A1 (A2 A3 ) + P (A1 (
Homework 2 Solution
2.84 Let B = A2 A3
P (A1 A2 A3 ) = P (A1 B) = P (A1 ) + P (B) P (A1 B);
(1)
First of all, using the additive law, we have
P (B) = P (A2 A3 ) = P (A2 ) + P (A3 ) P (A2 A3 ) = P (A2 ) + P (A3 ),
the last equality follows from the given c
Homework 1 Solution
2.1
A=cfw_FF; B=cfw_MM; C=cfw_MM,MF,FM;
A B = ;
A B=cfw_FF,MM;
A C = ;
A C=cfw_MF, FM, FF,MM;
B C = cfw_M M ;
B C=cfw_MM,MF,FM;
c
C B =cfw_MF,FM since B c = cfw_F F, M F, F M .
as the complement of B.
Note that we use B c for B
2
Homework 9 Solution
6.11
a. Let Y1 Exp(1) and Y2 Exp(1), we want to find the density function of U =
density function of Y1 and Y2 are
(
(
ey1 , y1 > 0
ey2 , y2 > 0
fY1 (y1 ) =
fY2 (y2 ) =
0,
y1 0
0,
y2 0
Y1 +Y2
.
2
The
And because Y1 and Y2 are independe
Homework 8 Solution
5.48 NOT independent since p(0, 0) 6= p1 (0)p2 (0)
5.52 Independent since the region is separable
5.56 Not independent since the region is NOT separable
5.58 Not independent since the region is NOT separable
5.70 Omitted.
P P
5.89 Cov(
m
~
m
.
_
m
w
20 Q>bdr>owm gm? Kw mm: 85 20 O>WQCF>HOWm
mow H
$6deva mm. meg
o 13%; $.75 mmo 695$. mun: 6.9%, $ng mm BESS?
o @302 <Odw OmW Enema Omrmwimm Bmmg. :ZO <OHA3
8% Ems: .B :20 WOHwamz.
u memgg pdmmdm $055 Um 3%: mm 8%an Hmjmhosmu Sea
amowBam ow
Mm) New, /, 6/43 i 30 ' @
J WIS ywz A fad/em g 3/ _ Kememkl Ar 74
E fejf fda I
1 of" 07Z /4 Pod W Pal/:3 7V4, 137 m le.
my Edi-Ski 575371ij 21' 78 Car; [Vb/1
Remember A %&/ 4 ya M I 4 M Ma.
you MW luv: 14/ J43 le) a; of fit/81w f4
6991;17:qu 0/ q 1417' MK
Chapter 3. Discrete random variables and probability
distributions.
Defn:
A :
random:
variable (r.v.) is a function that takes a sample point
:
in S and maps it to it a real number.
That is, a r.v. Y maps the sample space (its domain) to the real
number l
Chapter 6: Functions of Random Variables
We are often interested in a function of one or several random
variables, U(Y1 , . . . , Yn ).
We will study three methods for determining the distribution of a
function of a random variable:
1. The method of cdfs
Multivariate Distributions
In many applications we measure several r.v.s per individual.
Examples:
:
For a set of households, we can measure both income and the
number of children.
For a set of giraffes, we can measure their height and weight.
We can m
Continuous random variables
Continuous r.v.s take an uncountably infinite number of possible
values.
Examples:
:
Heights of people
Weights of apples
Diameters of bolts
Life lengths of light-bulbs
We cannot assign positive probabilities to every partic
Math 447 - May 13, 2013 - Final Exam Solutions
Name:
Read these instructions carefully: The points assigned are not meant to be a guide to the
difficulty of the problems. If the question is multiple choice, there is a penalty for wrong
answers, so that yo
Wee/nasaag OC'lOer 2] A C/A" #1:? @
gchea/v/e: 755/ z (rowan?) Nov M?
I!
Kecall A u 9 how: err" 641/ fer Mel. Salim.
7%; mm 4/1. .4 m; 7. AM c4, 7%. m; of
ood: givnla.
Tm 2 MW cove CA 9 2 I . 2 MW 2% WI:
72575, w MW id 7b v2 41 Itanwe M/e; M. %
Innolc Cov
Wedncrday Max/Mb ll Clef! 4 Z 5 @
KMMJMR 72: MW 20%? Yam/Ar 721 a Wain);
Corm/aak: 04 Im/e M' Ca va'mft a: a menu/we
_._.
0F how :02qu X W Y vary ivy/cfw_w i; [7 I
Arr," Jon X W V an: @494), cw. ,7 M
Cm/IMIM M01 X M Y ,J 1407 Ye/y Hwy,
gr exam/, If FavZX/