Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Solution to Exercise 2.28:
The purpose of this problem is to illustrate that for an arrival process with independent
but not identically distributed interarrival intervals, X1 , X2
M SS CHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2007
Final exam, 1:304:30pm, (180 mins/100 pts)
6.436J/15.085J
12/19/07
Problem 1: (24 points)
During the time interval [0, t], men and women arrive according to independent Poisson
processes with parameters 1 an
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Solution to Exercise 2.3:
a Given Sn = , we see that N (t) = n, for t only if there are no arrivals from to
t. Thus,
Pr (N (t) = n|Sn = ) = exp (
(t
)
b
Pr (N (t) = n) =
=
=
=
=
Z
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Solution to Exercise 2.28:
Suppose that the states are numbered so that state 1 to J1 are in the recurrent class 1,
J1 + 1 to J1 + J2 in recurrent class 2, etc. Thus, [P ] has the f
M SS CHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2008
Final exam, 1:304:30pm, (180 mins/100 pts)
6.436J/15.085J
12/18/08
Whenever asked to explain or justify an answer, a formal proof is not needed, but
just a brief explanation.
Problem 1: (30 points)
Consider
6 436/15 085
Midterm Exam
Date: October 23, 2006
Problem Which of the following functions is a distribution function? For those which are
compute the density function. For those which are not explain what fails.
.
F (x) =
1 e
0,
x2 ,
x 0;
otherwise.
B.
F
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Solution to Exercise 1.10:
a We know that Z(!) = X(!) + Y (!) for each ! 2 .
Pr(Z(!) = 1) = Prcfw_!; Z(!) = +1 or Z(!) =
!
= Prcfw_!; Z(!) = +1 + Prcfw_!; Z(!) =
1
Prcfw_!; Z(!) = +
Solutions to Homework
6.262 Discrete Stochastics Process
MIT Spring 2011
Solution to Exercise 1.3:
a Since A , A2 , . . . , are assumed to be disjoint, the third axiom of probability says that
1
X
[1
Am =
Pr
Pr Am
m=
m=
S1
Since = m= Am , the term on the
6.262 Discrete Stochastic Processes
MIT, Fall 2011
Monday April 4, 7:00-9:30pm, 2011
Solutions to Quiz
Problem 1: An innite sequence of packets are waiting to be sent, one after the other,
from point A to an intermediate point B and then on to C. The leng
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 1
Let cfw_Yn ; n 1 be a sequence of rvs and assume that limn!1 [|Yn |] = 0. Show that
cfw_Yn ; n 1 converges to 0 in probability. Hint 1: Look for the easy way. Hint 2: The