Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #13
a b
c d
space of M(2, 2).
1. Let W =
M(2, 2) d = a and c = b . Prove that W is a sub-
Proof: First, observe that the 2 2 zero matrix O =
0 0
0 0
is in W ; hence,
W is not empty.
a b
; so that
b a
Ne
Math 60. Rumbos
Fall 2014
Solutions to Assignment #17
1. Let f : R2 R2 be a function satisfying
f
1
0
2
, f
3
=
0
1
=
5
1
and f
1
1
=
3
.
2
(a) Show that f cannot be linear.
Solution: If f was linear, then we would have that
f
1
1
1
0
+
0
1
= f
= f
1
+f
0
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #14
a b
M(2, 2) d = a and c = b . It was shown in Probc d
lem 1 in Assignment #13 that C(2, 2) is a subspace of M(2, 2).
1. Let C(2, 2) =
(a) Prove that C(2, 2) = spancfw_I, J, where
1 0
0 1
I=
and J =
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #15
1. Let A be an m n matrix, and cfw_e1 , e2 , . . . , en denote the standard basis in Rn .
(a) Prove that Aej is the j th column of the matrix A.
R1
R2
Solution: Write A = . , where R1 , R2 , .
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #16
1. Dene T : R2 R2 as follows: For each v R2 , T (v) is the reection of the
point determined by the coordinates of v, relative to the standard basis in R2 ,
on the line y = x in R2 . That is, T (v) de
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #12
1
1
1. The vectors v1 = 1 , and v2 = 0 span a twodimensional subspace
2
1
3
in R ; in other words, a plane through the origin. Give two unit vectors which
are orthogonal to each other, and which al
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #19
1. Assume that T : Rn Rm is linear. Prove that T is onetoone if and only if
NT = cfw_0, where NT denotes the null space. or kernel, of T
Solution: Assume that T : Rn Rm is linear and that T is oneto
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #22
1. Let T : R2 R2 denote a linear transformation in R2 . Suppose that v1 and v2
are two eigenvectors of T corresponding to the eigenvalues 1 and 2 , respectively.
Prove that, if 1 = 2 , then the set c
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #21
1. Let R : R2 R2 denote rotation around the origin in the counterclockwise
through an angle . Let B = cfw_v1 , v2 , where
2
1
v1 =
and
1
.
2
v2 =
Give the matrix representation for R relative to B; t
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #20
1. In this problem and problems (2) and (3) you will be proving the Dimension
Theorem
dim(NT ) + dim(IT ) = n,
(1)
for a linear transformation T : Rn Rm .
Show that if NT = Rn , then T must be the ze
Department of Mathematics
Pomona College
Syllabus for Mathematics 60
Fall 2014
Time and Place:
MWF 11:00 am - 11:50 am MDSL 126
Instructor:
Dr. Adolfo J. Rumbos
Office:
MDSL 106
Phone / e-mail:
ext. 18713 / [email protected]
Office Hours:
appointment
MWF
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #18
1. Given two vectorvalued functions, T and R, from Rn to Rm , we can dene the
sum, T + R, of T and R by
(T + R)(v) = T (v) + R(v) for all v Rn .
(a) Verify that, if both T and R are linear, then so i
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #12
1. Compute the moment generating function, X (t), of a continuous random variable X with Uniform(1, 2) distribution. What should (0) be? Give also the
second moment and variance of X.
Solution: The
Math 151. Rumbos
Fall 2014
Solutions to Assignment #11
1. Let X Uniform(1, 2). Compute the variance of X.
Solution: The pdf of X is
1,
0,
fX (x) =
if 1 < x < 2;
elsewhere.
Then,
E(X) =
xfX (x) dx
2
x dx
=
1
=
3
,
2
and
x2 fX (x) dx
2
E(X ) =
2
x2 dx
=
1
=
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #22
1. Let X denote a random variable with mean and variance 2 . Use Chebyshevs
inequality to show that
Pr(|X |
k)
1
,
k2
for all k > 0.
Solution: By Chebyshevs inequality,
Pr(|X |
k)
1
2
= 2,
2
(k)
k
w
Math 151. Rumbos
Fall 2014
Solutions to Assignment #17
1. Let X and Y be independent Normal(0, 1) random variables.
Compute Pr(X 2 + Y 2 < 1).
Solution: Since X, Y Normal(0, 1), their pdfs are given by
1
2
ex /2 ,
fX (x) =
2
for x R,
and
1
2
fY (y) =
ey
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #20
1. Let X1 , X2 , X3 , . . . denote a sequence of independent, identically distributed
random variables with mean . Assume that the moment generating function
of X1 exists in some interval around 0.
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #16
1. Suppose that X Normal(, 2 ) and dene Z =
X
.
Prove that Z Normal(0, 1)
Solution: Since X Normal(, 2 ), its pdf is given by
fX (x) =
1
2
2
e(x) /2
2
for < x < .
We compute the cdf of Y :
FY (y)
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #18
1. We have seen in the lecture that if X has a Poisson distribution with parameter
> 0, then it has the pmf:
pX (k) =
k
e
k!
for k = 0, 1, 2, 3, . . . ; zero elsewhere.
Use the fact that the power
Math 151. Rumbos
Fall 2014
Solutions to Assignment #19
1. Let a denote a real number and Xa be a discrete random variable with pmf
pXa (x) =
1 if x = a;
0 elsewhere.
(a) Compute the cdf for Xa and sketch its graph.
(b) Compute the mgf for Xa and determine
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #15
1. Suppose X and Y are independent and let g1 (X) and g2 (Y ) be functions for
which E(g1 (X)g2 (Y ) exists. Show that
E(g1 (X)g2 (Y ) = E(g1 (X) E(g2 (Y )
Conclude therefore that if X and Y are ind
Department of Mathematics
Pomona College
Math 151. Probability
Fall 2014
Course Outline
Time and Place:
MWF 9:00 am 9:50 am
Seaver Commons 102
Instructor:
Dr. Adolfo J. Rumbos
Office:
Mudd Science Library 106
Phone/e-mail:
ext. 18713 / [email protected]
Math 151. Rumbos
Fall 2014
1
Solutions to Exam 3 (Part I)
1. Let X1 , X2 , X3 . . . denote a sequence of random variables.
(a) State the Central Limit Theorem in the context of the sequence (Xk ).
Answer: Let (Xk ) be a sequence of independent, identicall
Math 151. Rumbos
Fall 2014
1
Solutions to Exam 3 (Part II)
1. A company manufactures a brand of incandescent light bulbs. Assume that the
light bulbs have a lifetime in months that is normally distributed with mean 3.5
and variance 1; assume also that the
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #23
1. Let X be continuous random variable with E(|X|) < . Derive the following
version of Markovs inequality: For every > 0,
Pr(|X|
E(|X|)
.
)
(1)
Solution: Let fX denote the pdf of X and compute
|x|fX
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #21
1. An experiment consists of rolling a die 81 times and computing the average
of the numbers on the top face of the die. Estimate the probability that the
sample mean will be less than 3.
Solution:
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #14
1. Suppose that in an electric display sign there are three light bulbs in the rst
row and four light bulbs in the second row. Let X denote the number of bulbs
in the rst row that will be burned out
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #13
1. Let X Gemetric(p), where 0 < p < 1. Compute the mgf of X and use it to
compute the E(X), E(X 2 ) and var(X).
Note: You will need the fact that
ak =
k=1
a
,
1a
for |a| < 1.
(1)
Solution: Let X Gem
Math 151.
ProbabilityRumbos
Spring 2014
Topics for Exam 1
1. Probability Spaces
1.1. Sample spaces
1.2. Fields
1.3. Probability functions
1.4. Independent events
1.5. Conditional probability
2. Random Variables
2.1. Continuous and discrete random variable
Math 151.
ProbabilityRumbos
Fall 2014
Topics for Exam 2
1. Expectations of Random Variables
1.1. Expected Value a random variable
1.2. Expected value of functions of random variables
1.3. Moments, variance and and moment generating function
1.4. Uniquenes