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
Math 151. Rumbos
Fall 2014
1
Topics for Final Exam
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 variables
2.2 Cumulat