IEOR 4102, HMWK 2 Solutions, Professor Sigman
1. Consider the Rat in the Open Maze; 4 rooms, and the outside (state 0), but now the
probabilities are different whenever the rat leaves room 3: P3,1 = 1/8, P3,4 = 7/8;
all the other probabilities are equally
IEOR 4102, HMWK 1, Professor Sigman
1. An asset price starts off initially at price $7.00 at the end of a day (day 0), and at
the end of each consecutive day, independent of the past, the price goes up by one
dollar (with probability p = 0.6) or down by o
IEOR 4102, HMWK 3, Professor Sigman
1. A stock has an initial price of S0 = 40. Sn denotes the price at time n, where we assume
the binomial lattice model with parameters
u = 1.25
d = 1.01
p = 0.55.
(a) Compute E(S1 ) and E(S2 ).
SOLUTION: Note that E(Y )
e0 i2= cfw_H,Tr
:; i lo ,wr
rvor irv A
@ uB) c = ,4() Be
- ft PRoBl!B(t.irr fi,(.Nc.,-fo,v p 6N
irv &tfC .
)i: = A uB C
c 2016 by Karl Sigman
Gamblers Ruin Problem
Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial
fortune of $i and then on each successive gamble either wins $1 or loses $1 independent of the
past with probabil
c 2016 by Karl Sigman
Stopping Times: Definition
Given a stochastic process X = cfw_Xn : n 0, a random time is a discrete random variable
on the same probability space as X, taking values in the time set N = cfw_0, 1, 2, .
Solution to Midterm Exam (II)
Question 1 (10 2 =20 pts) Find the limit if it exists, or show that the limit does not exist.
x6 +y 2
x2 +y 2 +11
(a) The limit does not exist. For example, along the x-a
260 Introduction to Probability Models
To recursively solve for Vk (j), use that
Vk(j) = max max Pcfw_Xk2 = (i1, . . . , ik_2), Xk_1 = i, Xk = j, S" = 5k
l 11 . lk2
. . . _1
= ma)? max Pcfw_Xk2 = (11,.-.,lk2),Xk1 =1,Sk = Sic1,
l 11 . zk_2
Xk = 135k = Sk
Dr. A. B. Dieker
Introduction to Probability and Statistics
due on Wednesday March 23, 2:40pm
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.g., on wik
L? EYMuIasQ-mh, G: (V:E\
g. G-iEarefjiis s ISl'| V SS V. Saviour 4.0me G t;
535 = 0-1 \
4? > \N 2. F75 2 Vgsv <4,V o u
l ' (met; ,2 31>
3 5 j t a .0
2. hn 26.3 35 de; @395 WA 1440+ 0 35
fans; cfw_2 = 1 u CL C!
-Z(t.-t)eE cfw_151" " H eff),
IEOR 4102, Midterm Exam, Spring 2016. 75 Minutes.
110 Points Total. Professor K. Sigman
Open Notes (anything on the course website plus your notes from class), but no books and
no electronic devices of any kind.
1. (40 points, 10 each) You arrive at the W