ma3160a4 ma5160a4 ma5610a4 0910B
The Poisson Process
1. Cars pass a certain street location according to a Poisson process with rate . A man
to cross the street at that location waits until he can see
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Random Variables
3
. The process
5
continues
until a player has won two more games than the other . The overall winner is the first
player to have
won two more games t
ma3160 Test , 2003-2004B
1.
Two binary communication systems A, B take turns to transmit 0 or 1 to some receiver, they do this
3
independently of the other in the order ABAB . The probability that sys
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Markov Chains
1. Three white and three black balls are distributed in two urns in such a way that each contains three
balls. We say that the system is in state i , i =
ma3160a6 ma5160a6 ma5610a6 0910B
Markov Chains
1. Three white and three black balls are distributed in two urns in such a way that each contains three
balls. We say that the system is in state i , i =
ma3160a5 ma5160a5 ma5610a5 0910B
The Poisson Process
1. Cars pass a certain street location according to a Poisson process with rate . A man who wants
to cross the street at that location waits until
ma3160a5 ma5160a5 ma5610a5 0910B
The Poisson Process
1.
Cars pass a certain street location according to a Poisson process with rate . A man who wants to
cross the street at that location waits until
MA3160 Random Vector 1112B
1 Jointly Distributed Random Variables
1.1 Let ( , , P ) be a probability space. Let X i : R , i = 1,
X = ( X1, X 2 ,
, n be random variables.
, X n ) is called a random vec
ma3160a4 ma5160a4 ma5610a4 0910B
1.
Distributions
Let X and Y be independent Bernoulli random variables with parameter
1
. Show that X + Y and
2
X + Y are dependent though uncorrelated.
Proof:
01
01
X
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Random Vectors
1. Express the distribution of the followings in terms of the distribution function FX of the random
variable X.
(a) X + = maxcfw_ 0, X .
(b) X = mincf
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Probability
1. Consider two events A and B such that P ( A ) = 0.4 and P ( B ) = 0.7 . Determine the maximum and
minimum possible values of P ( A B ) and the condition
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1.
Queueing
Cars arrive at a petrol pump with exponential inter-arrival times having mean
1
minutes. The
2
1
minutes per car to supply petrol, the service times being
ma3160a4 ma5160a4 ma5610a4 0910B
1.
Distributions
Let X and Y be independent Bernoulli random variables with parameter
1
. Show that X + Y and
2
X + Y are dependent though uncorrelated.
2. Let X 1 , X
ma3160a3 ma5160a3 ma5610a3 0910B
Random Vectors
1. Express the distribution of the followings in terms of the distribution function FX of the random
variable X.
(a) X + = maxcfw_ 0, X .
(b) X = mincf
ma3160a2 ma5160a2 ma5610a2 0910B
Random Variables
3
. The process
5
continues
until a player has won two more games than the other . The overall winner is the first
player to have
won two more games t
ma3160a1 ma5160a1 ma5610a1 0910B
Probability
1. Consider two events A and B such that P ( A ) = 0.4 and P ( B ) = 0.7 . Determine the maximum and
minimum possible values of P ( A B ) and the condition
ma3160a4 ma5160a4 ma5610a4
1.
0910B
Distributions
Let X and Y be independent Bernoulli random variables with parameter
1
. Show that X + Y and
2
X + Y are dependent though uncorrelated.
Proof:
1
1
Cov
Hints of Take-home Quiz I
Please submit the solution of Quiz 1 in person or by inserting it into the opening on my office door
(Y6506) on or before February 4. The quiz is worth 6 marks.
Question 1
3
Take-home Quiz II
Please submit the solution of Quiz 1 in person or by inserting it into the opening on my office door
(Y6506) on or before March 18. The quiz is worth 8 marks.
Question
At a party n n
Take-home Quiz III
Please submit the solution of Quiz III in person on April 1 or by inserting it into the opening on my office
door (Y6506) on or before April 1. Try to attempt Question 1, the questi
MA3160 Random Variable 1112B
1 Random Variable
1.1 Let ( , , P ) be a probability space. Let X be a function from to R, that is, X : R , where R
is the set of real numbers. X is said to be a random va
Take-home Quiz IV
Please submit the solution of Quiz IV in person on April 22 or by inserting it into the opening on my office
door (Y6506) on or before April 22.
Students of MA3160, MA5160 attempt Qu
Take-home Quiz III
Question 1
Customers can be served by any of three servers where the service times of server i are independently
exponentially distributed with rate i , i = 1, 2,3 . Whenever a serv
Take-home Quiz II
Please submit the solution of Quiz 1 in person or by inserting it into the opening on my office door
(Y6506) on or before March 18. The quiz is worth 8 marks.
Question
At a party n n
Take-home Quiz I
Question 1
An urn contains n red balls and m blue balls, where nm 0 . Balls are removed at random and discarded
until the first time a ball ( B , say) is removed having a different co
Take-home Quiz IV
Please submit the solution of Quiz IV in person on April 22 or by inserting it into the opening on my office
door (Y6506) on or before April 22.
Students of MA3160, MA5160 attempt Qu
Take-home Quiz III
Please submit the solution of Quiz III in person on April 1 or by inserting it into the opening on my office
door (Y6506) on or before April 1. Try to attempt Question 1, the questi
Take-home Quiz II
Please submit the solution of Quiz 1 in person or by inserting it into the opening on my office door
(Y6506) on or before March 18. The quiz is worth 8 marks.
Question
At a party n n
Take-home Quiz I
Please submit the solution of Quiz 1 in person or by inserting it into the opening on my office door
(Y6506) on or before February 4. The quiz is worth 6 marks.
Question 1
3 marks
An
ma3160a3 ma5160a3 ma5610a3 0910B
Random Vectors
1. Express the distribution of the followings in terms of the distribution function FX of the random
variable X.
(a) X + = maxcfw_ 0, X .
(b) X = mincf