State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) If P is a transition matrix of a Markov Chain, then all the rows of are identical.
b) In a variance-covariance matrix all elements are always positive.
c) If X1, X2, X3 are iid from , then
follows
.
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function .
a) False. A transition matrix of a Markov chain has rows that represent conditional probability distributions of moving from a given state to other states. Each row must sum to 1, but the rows need not be identical. For example, in a two-state Markov __________ _________ __________ __________ ________.
_________ _____ __________ ___ ___ _______ ______ __________ _____ ____.
________ ______ ______ _________ ________ _____.
____ ______ _______ ____ _______ ___ ___.
___ _______ _______ _____ _______ ___ ___ ______.
_____ ______ ____ __________ _______ __________ _________ _____.
_________ __________ _____ ___ ________ _________ _____ ____ ____ ___ _________.
____ _______ _____ _________ _________ _________ ____ _________ ________ ______ _____.
____ _______ _________ __________ ___ _____.
___ ____ __________ _________ _______ _______ ____.
_________ _______ ____ _________ ____ _________ _____ _______ _________.
__________ __________ ______ _______ _______ ______ ___ ____.
________ ___ __________ ________ ___ ____.
________ __________ ______ ________ ___ _____ ____ _______ _________.
________ ________ __________ ________ ___ ___ ______ _______ _____ ___.
_____ _____ ___ __________ _________ ____ ______ ___ ______.
_____ _____ _____ _______ _______ ____ __________ ________ _____ _______ ______.
________ _________ ________ ______ ________ ______ ___ ______ ________ _______.
__________ ________ _________ ______ ______.
__________ __________ _______ _____ _______ _________ _______.
_________ ____ ___ _________ ________ _________ ______.
__________ ____ ___ __________ __________ _____ _______.
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Let be a normal random vector with the mean vector
and covariance matrix
. Suppose
, where
and
.
i) Find .
ii) Compute E(Y).
iii) Find the covariance matrix of Y.
iv) Find .
At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
i) Probability that there is no customer at the counter.
ii) Probability that there are more than two customers at the counter.
iii) Average number of customers in a queue waiting for service.
iv) Expected waiting time of a customer in the system.
i) Probability that a customer wait for 0.11 minutes in a queue.
State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) If P is a transition matrix of a Markov Chain, then all the rows of are identical.
b) In a variance-covariance matrix all elements are always positive.
c) If X1, X2, X3 are iid from , then
follows
.
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function .
A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval [2, 5] years. Further, planned replacements take place every 3 years.
Compute
i) long-terms rate of replacements.
If the random vector Z be , where:
and
Find r34, r34.21.ii) long-terms rate of failures.
Suppose life times are i.i.d. uniformly distributed on (0, 3) and
and
. Find:
i)
ii) T which minimizes C(T) and which is the better policy in the long-run in terms of cost.
A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval [2, 5] years. Further, planned replacements take place every 3 years.
Compute
i) long-terms rate of replacements.
ii) long-terms rate of failures.
A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined:
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
Find and
.