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.
To determine whether the Markov chain is irreducible, we need to check if it is possible to reach any state from any other state. If the chain is irreducible, we proceed to classify the states into recurrent, transient, periodic, and find the mean recurrence time.
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Let ( ,X Y) have the joint p.d.f. given by:
f(x,y)=
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 Y = .y
iv) Compute (E X | Y = )y and (E Y | X = ).x
A random sample of 12 factories was conducted for the pairs of observations on sales (X1) and demands ( X2) and the following information was obtained:
The expected mean vector and variance covariance matrix for the factories in the population are:
=
and
Test whether the sample confirms its truthness of mean vector at 5% level of
significance, if:
i) Σ is known,
ii) Σ is unknown.
Let the joint probability density function of two discrete random X and Y be given as:
i) Find the marginal distribution of X and Y.
ii) Find the conditional distribution of X given Y=1.
iii) Test the independence of variable s X and Y.
iv) Find V
Determine the principal components 1 Y2 Y , and Y3 for the covariance matrix:
Also calculate the proportion of total population variance for the first principal component.
Let the random vector X =(X1,X2,X3) has mean vector [− ,2 ,3,4 ] and variance covariance matrix = Fit the equation Y = b0+ b1 X+ b2 X 2 Also obtain the multiple correlation coefficient between X3 and [X ,X2 ]
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.
v) Probability that a customer wait for 0.11 minutes 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: (5)
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
Find (P C|A ), (P |B C), (P A ∩ C|)B ), (P A), (P )B and (P A∩ ).B