CS112 Homework 7
Q1. Consider a birth-death system in which k = and k = k for k 0. For all k, nd
the dierential dierence equations for Pk (t) = P [k in system at time t].
Q2. Consider a taxi station where taxis and customers arrive in a poisson process wi
CS112: Modeling Uncertainty in Information Systems
Homework 5
Due Friday, June 8, at the beginning of section
Please refer to the course academic integrity policy for collaboration rules. In particular, be sure to include
a list of anyone with whom you ha
CS112 HOMEWORK 3
1. An articial intelligent robot is detecting an item placed in front of him using a newly
invented computer vision algorithm. After calculating, he thinks that the item has 35%
probability to be an object A, 15% probability to be an obje
CS112 DISCUSSION 2
1. Probability Basic
(1) Let E, F, G be three events. Find expressions for the events that of E, F, G
(i) only E occurs
E (F G)c
(ii) both E and F but not G occur
E F Gc
(iii) at least one event occurs
EF G
(iv) at least two events occu
CS112 HOMEWORK 1
1. Find the Laplace Transform of:
(1) f (x) = eax
(2) f (x) = x2
(3) f (x) = 4x2 3x + 7
(4) f (x) = (x 1)2
2. Sum the series:
n
3
i=1 i
3. Answer the following questions:
(1) There are 6 balls with dierent colors and 3 dierent boxes. You
CS112 HOMEWORK 4
1. A given program has an execution time T that is uniformly distributed between 10 and 20
seconds. The number of interrupts that occur during execution is a Poisson random variable
with parameter t where T = t is the program execution ti
CS112 HOMEWORK 1
1. Combinatorics
(1) For years, telephone area codes in the United States and Canada consisted of a
sequence of three digits. The first digit was an integer between 2 and 9, the second
digit was either 0 or 1, and the third digit was any
CS112 Homework 6
1. Consider a birth-death process with 4 states, where the transition rate from state 2 to state 1
is q21 = , and q23 = . Show that the mean time spent in state 2 is 1/( + ).
Solution:
Suppose the system has just arrived at state 2. The t
CS112 HOMEWORK 4
1. The packets are sent from a server to a client host. The client host notices that the time
interval between two consecutive packets is uniformly distributed between 10ms to 20ms.
(1) Suppose one packet just arrived. What is the probabi
Homework 5
1. A given program has an execution time that is uniformly distributed between 10 and 20
seconds. The number of interrupts that occur during execution is a Poisson random variable with
parameter t where t is the program execution time. The prob
CS112 - Homework 6 (Part 1)
1. There are N machines where N is a large number. Each machine has one of
three possible states and changes states (independently) according to a Markov
Chain with transition probabilities
0.7 0.2 0.1
0.2 0.6 0.2
0.1 0.4 0.5
CS112 Discussion 7
1
Stochastic Process
Three important characteristics
State space(discrete or continuous)
time parameter(discrete or continuous)
Relation between cfw_Xt (w)(dependence or independence)
2
Irreducible and Periodicity
All states communic
CS112 DISCUSSION 2
1. Probability Basic
(1) Let E, F, G be three events. Find expressions for the events that of E, F, G
(i) only E occurs
E (F G)c
(ii) both E and F but not G occur
E F Gc
(iii) at least one event occurs
EF G
(iv) at least two events occu
Simple Properties of Z-Transforms
Property
Sequence
z-transform
1.
linearity
c xn + d yn
c Z[xn ] + d Z[yn ]
2.
delayed unit step
u[n m]
z 1m
z1
3.
single delay
xn1 u[n 1]
1
Z[xn ]
z
4.
time delayed shift
xnm u[n m]
1
Z[xn ]
zm
5.
time advance
xn+m
z m (Z
CS112 HOMEWORK 2
1. Express each of the following events in terms of the events A, B and C as well as the
operations of complementation, union and intersection:
(a) at least one of the events A, B, C occurs
(b) at most one of the events A, B, C occurs
(c)
Christopher Zhu
UID:104455996
CS112
Homework 5
Problem 1.
There are N machines where N is a large number. Each machine has one of three possible states and
changes states (independently) according to a Markov Chain with transition probabilities [0.7 0.2 0
Properties of Laplace-Transforms
cfw_sourcez Electric Circuit (10h Ed by Nilsson and Riedel)
TABLE 12.1 An Abbreviated List of Laplace Transfonn Pairs
Tyin- no (I > 0) Rs)
(impulse) 8(1) 1
I
(step) 11(1) ;
(rim ) I l
. p A.2
(exponential) e'" l
.r +
CS 112 Final
Summer 2005
Name:
Student ID:
1: Closed book, closed notes. One page cheat sheet is allowed. Calculators, PDAs, or Laptops
are not allowed.
2: Show all the work. In case of doubts, state any assumption you make.
Problem 1 (20 Points)
Problem
CS112 DISCUSSION 1
1. Elementary Math
(1) If a1 = 1 , an+1 = an +
2
an+1 an =
1
n
an an1 =
1
n1
what is an ?
1
n+1
1
n
1
n2
an1 an2 =
1
,
n2 +n
1
n1
1
3
a2 a 1 = 1
1
2
a3 a 2 =
1
2
1
So: an a1 = 1 n , an =
3
2
1
n
(2) If a1 = 1, an+1 = 2an + 3, what is a
CS112 DISCUSSION 4
1. Uniform Distribution
The amount of time, in minutes, that a person must wait for a bus is uniformly distributed
between 0 and 15 minutes, inclusive.
(a) What is the probability that a person waits fewer than 12.5 minutes?
Let X = the
Discussion 5
1
Review
1.1
Expections
For a discrete random variable X with pmf pX (x),
E[X] =
xpX (x)
x
For a continuous random variable X with pdf f (x),
xf (x)dx = X
E[X] =
Binomial distribution: N p
1
Geometric distribution: p
Poisson distribution:
Un
CS112 Discussion 3 Solutions
1. Suppose that two cards are drawn from an ordinary deck (52 cards), one-by-one, at random
and with replacement (the drawn card is put back into the deck). Let X be the number of
spades drawn. If an outcome of spades is denot
CS112 DISCUSSION 6
1
Review
1. stochastic process: a family of random variable indexed by time
2. Three important characteristics of a stochastic process
(a) State space (discrete or continuous)
(b) Time parameter (discrete or continuous)
(c) Relation bet