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.
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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.
Determine the principal components Y1, Y2 and Y3 for the covariance matrix:
Also calculate the proportion of total population variance for the first principal component.
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 level of significance, if:
i) is known,
ii) is unknown.
[You may use: ]
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 (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
.
Consider the Markov chain with three states, S = ,{1 ,2, 3} following the transition matrix
i) Draw the state transition diagram for this chain.
ii) If , then find
.
iii) Check whether the chain is irreducible and a periodic.
iv) Find the stationary distribution for the chain.
Let the random vector has mean vector [-2, 3, 4] and variance
covariance matrix . Fit the equation
. Also obtain the multiple correlation coefficient between X3 and [X1, X2].
b) Define ultimate extinction in a branching process. Let
0 < b < c < b + c < 1 and . Then discuss the probability of extinction in different cases for
or E(X1) < 1.