m/O {OL‘IA‘OM
Problem 1 (35 pts)
At the end of a month, a large retail store classiﬁes each of its customer’s accounts according to current
(0), 30—60 days overdue (1), 60—90 days overdue (2), more than 90 days (3). Their experience indicates
that the a
STAT/MATH 416
Spring 2016
Homework #1
Note: For this and future homework assignments, the chapters and respective questions
are from Introduction to Probability Models by Sheldon Ross, 11th edition. If you have
another edition, first check that the proble
MrptmM 2 5me
1. (20 pts) An auto insurance company classies its customers in four categories: excellent (0), good (1),
satisfactory (2), and poor (3). Let Xn be a customers category for the nth year.
'0 l 2. 3
a) From experience the categories can be mo
Solution to HW1
1. (a) F \ E c \ Gc
(b) E \ F \ Gc
(c) E [ F [ G
(d) (E \ F ) [ (E \ G) [ (F \ G)
(e) E \ F \ G
(f) E c \ F c \ Gc
(g) (E c [ F c ) \ (E c [ Gc ) \ (Gc [ F c )
(h) (E \ F \ G)c
2.
P =
(
i 1
36
13 i
36
2i7,
7 < i 12 .
3. A= event that both
Solutions to HW4
February 13, 2017
70
(t) = E[eXt ] =
X
etk (1 p)k1 p
k=1
p X t
=
[e (1 p)]k
1 p k=1
p
(1 p)et
1 p 1 (1 p)et
pet
=
1 (1 p)et
=
where we have assumed that (1 p)et < 1, i.e.
T = cfw_t R : t < log(1/(1 p)
72
Let Xi , i = 1, 2, denote the mont
Solutions to HW9
April 1, 2017
4.70
a).
i i
i2
= 2,
m m
m
mi mi
(m i)2
Pi,i+1 =
=
m
m
m2
2mi 2i2
Pi,i = 1 Pi,i1 Pi,i+1 =
m2
Pi,i1 =
b).
i =
m
i
m
mi
2m
m
i = 0, 1, 2 m
,
c).
Firstly, we verify the assumption in b) is correct. Since the MC is irreducible,
STAT/MATH 416
Spring 2017
Homework #6
Chapter 3: # 36, 37, 38, 44, 49[BONUS], 51, 56
Problem A:
A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right,
then it will wander around in the maze for three minutes
STAT/MATH 416
Spring 2017
Homework #4
Chapter 2: # 70, 72, 77, 78, 80, 86
Note: In #70 you need to derive (!) the formula for the moment generating function.
Problem A:
Let 1 , , be independent random variables and ~( , ), = 1, , . Use the
moment generati
STAT/MATH 416
Spring 2017
Homework #11
Chapter 5: # 57, 83[BONUS]
Problem A:
Customers arrive at a bank according to a Poisson process with rate 10 per hour. Provided two customers
arrive in the first 5 minutes, find the probability that:
a) both arrived
STAT/MATH 416
Spring 2017
Homework #3
Chapter 2: # 54, 55(a,b), 55(c)[BONUS], 60, 65, 68
Problem A:
Let the joint pmf of and be (, ) = 2 , = 1,2,3, = 1,2.
a) Find constant .
b) Find the marginal pmf of and the marginal pmf of .
c) Are and independent? Jus
STAT/MATH 416
Spring 2017
Homework #7
Note: In this homework, except for the bonus problem, operations with matrices/vectors
are to be done by hand (use calculator only for basic arithmetic operations).
Problem A:
A Markov chain 0 , 1 , 2 , with state spa
STAT/MATH 416
Spring 2017
Homework #1
Note: For this and future homework assignments, the chapters and respective questions
are from Introduction to Probability Models by Sheldon Ross, 11th edition. If you have
another edition, first check that the proble
5.1mm +0 Hw Ia
n;.mwi+k +hc if u pwple in H: sho) as He s'hJa , we 32% a bir+h M4 lu' PM" W
A5: 3, 10.! . 13:4, j=l.z
grc34n, WW4?!
Sh P'+"'P-'-" he kg": P=% l Y=%l ?;";11
h'-' V'P'*|Pt+2-Ps= '3'? J
(5) Th Phrur'hon of partial cus'lmm enhrinJ +Le ska? ix
Quiz 1 Solutions
1. The number of years a radio functions is exponentially distributed with the mean
of 7 years. If you buy a previously owned radio that has been in use for 3 years,
what is the probability that it will be working after additional 8 years
Quiz 4 Solutions
1. (a) A system consists of three components whose lifetimes (in hours) are independent
and exponentially distributed with means 2, 5, and 4, respectively. The system
works as long as all three components work. What is the probability tha
Quiz 2 Solutions
1. Find E(X) using whichever parts of the following that are applicable: fX (x) = 1, 0 <
x < 1, fY (y) = ln y, 0 < y < 1, E(X|Y = y) = y1
, 0 < y < 1, E(Y |X = x) = x2 ,
ln y
0 < x < 1.
R
Using the Law of Iterated Expectation, E(X) = E(E(
Solutions to Practice Problems for Midterm 2
1. (a), (c), and (d) cannot be tpm:
(a) - is not a square matrix
(c) - contains entries that cant be probabilities (2, -1)
(d) - rows do not sum up to 1
(b) is a square matrix, all entries are between 0 and 1,
Practice Problems for Midterm 2
1. Determine which of the following matrices are valid t.p.m. and why:
2. Consider the Markov chain with states 0, 1, , 6 and transition probability matrix
0.7 0
0
0.1 0.2 0.3
0
0 0.5
0
0
0
0.6 0
0
0
0
0
0
0
0
For each stat
Mia/Mm / Soho-519M
l. (9 pts) The number of cars passing a remote intersection per hour has Poisson distribution with mean 2.
The number of cars passing in an hour is independent of the number of cars passing in any other hour.
:1) Find the probability
Practice Problems for Midterm 1
1. Batches that consist of 15 coil springs from a production process are checked for conformance to
customer requirements. The mean number of nonconforming coil springs in a batch is 3. Assume that
conformance statuses of d
Solutions to Practice Problems for Midterm 1
1. Here we have a Binomial experiment:
fixed # of springs (trials), = 15
they are independent
each spring is either non-conforming (success) or not
probability of non-conformance is the same for all springs