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.
To express a factor model for three random variables \(X_1, X_2, X_3\) with one common factor, we can use the following factor model:
\[X_i = \alpha_i F + \epsilon_i\]
where:
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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.
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 P (X1=1) P(X1 =2)=1/4 then find P (X1=3,X2=2,X3=1)
iii) Check whether the chain is irreducible and a periodic.
iv) Find the stationary distribution for the chain.
If the random vector Z be N ( , ), 4 µ Σ where:
and
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.
Suppose life times X ,X ,..... 1 2 are i.i.d. uniformly distributed on (0,3)and C1 = 2 and C2 = .8 .Find:
i) µ T
ii) T which minimizes C (T) and which is the better policy in the long-run in terms of cost.
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 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