Consider a Markov chain with transition probability matrix:
i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.
ii) Find the limiting probability vector.
## Given transition matrix [ P=\begin{pmatrix} 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ \frac14 & \frac18 & \frac18 & \frac12 \end{pmatrix} ] States: ( {1,2,3,4} ) # (i) Irreducibility and classification of states ## Irreducibility * (1 \to 3 \to 2 \to 4) * From state 4, transitions are possible to all states (including itself). Hence every state can be reached from ______ _______ _____ _____ ______ ___ ________ ______ _____ _______ _________.
________ __________ ____ ________ ______.
___ ___ ______ ________ _______.
__________ ______ ________ _____ __________ _____ _______ ___ ________.
____ _____ _________ ____ ____ ____ ____ _________ _______ _____.
_______ ___ _________ _______ _____ ____ _______.
_________ _________ __________ ________ ________ ________ _________ __________ ________ ___ ___ __________.
_________ ______ _________ ______ _________.
______ ____ __________ ____ __________.
____ _____ _____ _____ __________.
________ _________ ____ ____ __________.
_______ _____ _________ __________ __________ _______ _____ ______ ___.
_______ ___ ___ ____ _____ __________ _________ _________ ____ _____ _______.
______ _____ ______ ____ _____ ______ __________ ____ ____ ________ ______ _______.
_______ ___ _________ ________ ________.
_____ ___ ________ _____ ___ _______ _____ _____ ___ _____ __________.
____ ___ ___ _____ _____ __________.
________ _______ _________ ____ ___ ____ _________.
________ _________ ________ ______ _______ _________ ________ _________.
____ ______ _______ _________ _________ _________ ___ _______ _________ ____ _______ ______.
_____ _______ _____ _____ ______ _________ _____ ______ _______ ___ ___ ______.
____ _____ ___ ___ _____ ___ ________.
____ ______ ____ _____ _______ _______.
__________ ____ __________ _____ ______ ________ ___.
_____ _________ ___ ______ ____ ______ _________.
________ ___ _______ ____ _______ _______ _____ ___.
__________ ___ _______ ___ ________ ________ ________ ____ _______ __________ ____.
______ _______ _________ ____ ______.
__________ _________ ______ __________ __________ _____ ______ _______ _________ _______ _________.
___ ______ ________ __________ _______ ______ _______ ______ ____.
_________ ____ ________ __________ _______ __________ ___ ____ __________ ____ ____ _________.
__________ _____ _____ _____ _____.
____ _________ ________ _____ ________ ____ ___.
_____ _______ _______ _____ ___ ___ _______ __________ ________ ___.
__________ _______ _______ ___ ________ ________ ____ ______ ____.
_________ __________ __________ ____ ______ __________ ___ ______ ____ __________ ____.
_____ ______ _________ ______ _____ _____ __________.
Get Full Answer on WhatsApp
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 .
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 .
A service station has 5 mechanics each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at a service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson and service times are exponentially distributed, determine:
i) Identify the model.
ii) The probability that the system shall be idle.
iii) The probability that there shall be 3 scooters in the service centre.
iv) The expected number of scooters waiting in a queue.
v) The expected number of scooters in the service centre.
vi) The average waiting time in a queue.
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
.
If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where
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.
Let (X, Y) have the joint p.d.f. given by:
i) Find the marginal p.d.f.'s of X and Y.
ii) Test the independence of X and Y.
iii) Find the conditional distribution of X given .
iv) Compute and
.