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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 than the other.
(a) Find the probability that A gets the
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 that no cars will come by in the
time
units.
(a) Find
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 system A transmits a 1 is
5
2
while the probability that s
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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 = 0,1, 2,3 , if the first urn contains i white balls. At
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 = 0,1, 2,3 , if the first urn contains i white balls. At
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 vector of dimension n.
The joint cumulative distribution f
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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 he can see that no cars will come by in the
next T
time
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 he can see that no cars will come by in the next T time
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
Y
1 1,
1 1.
Probability
Probability
22
22
Consider 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 = mincfw_ 0, X .
+
(c) X = X + X .
(d) X .
Solution:
(a)
Let
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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 conditions under which each of the value is
attained.
Solution:
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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
3
exponentially distributed. Determine
(a) the average
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 2 be independent random variables with Poisson distrib
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 = mincfw_ 0, X .
+
(c) X = X + X .
(d) X .
2. Let X be a Bern
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 than the other.
(a) Find the probability that A gets the
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 conditions under which each of the value is
attained.
2.
3.
(
Le
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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 ( X + Y , X Y ) = E ( X + Y ) X Y E ( X + Y ) E ( X Y
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 marks
An urn contains n red balls and m blue balls, whe
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 numbered men take off their hats. The hats are then mixe
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 question is worth 5 marks. If you also
attempt Question 2, 2
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 variable if for every real number a,
cfw_ : X ( ) a , the
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 Question 1 and the question is worth 6 marks.
Students of
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 server becomes free, the customer who has
been waiting the
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 numbered men take off their hats. The hats are then mixe
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 colour from its predecessor. The ball B is
now replaced a
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 Question 1 and the question is worth 6 marks.
Students of
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 question is worth 5 marks. If you also
attempt Question 2, 2
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 numbered men take off their hats. The hats are then mixe
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 urn contains n red balls and m blue balls, where nm 0 .
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 = mincfw_ 0, X .
+
(c) X = X + X .
(d) X .
Solution:
(a)
Let