Consider three random variables X1, X2, X3 having the covariance matrix
Write the factor model, if number of variables and number of factors are 3 and 1 respectively.
The given covariance (correlation) matrix is [ R=\begin{bmatrix} 1 & 0.12 & 0.08\ 0.12 & 1 & 0.06\ 0.08 & 0.06 & 1 \end{bmatrix}. ] For a one–factor model, (R=\Lambda\Lambda' + \Psi), where (\Lambda=(\ell_1,\ell_2,\ell_3)') ________ _____ __________ __________ __________ ________ ___ ____ _________.
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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 .
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.
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.
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
.
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 .
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.