MA3160a6 Markov Chain 1112B
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 each
step, we draw
MA3160a5 Poisson Process 1112B
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
units.
(a) Find the p
MA3160a4 Distribution 1112B
1.
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.
3.
Let X 1 , X 2 be independent random variables with Poisson distributions Poisson (
MA3160a3 Random Vector 1112B
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)
(c)
X = mincfw_0, X .
X = X+ + X.
2.
(d) X .
Let X be a Bernoulli random variable, so
MA3160a2 Random Variable 1112B
1.
2.
3.
4.
3
. The process continues
5
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 overall
MA3160a1 Probability 1112B
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.
Let A, B, C be events,
MA3160a7 1112B
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 number of cars wai
MA3160a5 Poisson Process 1112B
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
units.
(a) Find the p
MA3160a4 Distribution 1112B
1.
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:
0
X
1
Probability
2
Consider X + Y
1
0
Y
,
1
1
Probability
2
2
.
1
1.
2
111
=.
22
MA3160a3 Random Vectors 1112B
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 F+ ( x ) = P (X + x )
MA3160a2
1.
Random Variable 1112B
3
. The process continues
5
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 overall winner w
MA3160a1 Probability 1112B
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:
P( A B) = P( A) + P
MA3160a7 Queueing 1112B
1.
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 number of cars wai
MA3160a6 Markov Chain 1112B
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 each
step, we draw