State whether the following statements are True or False. Justify your

Question

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 lim Pⁿare identical.

b) In a variance-covariance matrix all elements are always positive.

c) If X_1, X₂ ,X₃ , are iid from N₂ ( µ , $\sum$ ), then $\frac{X_{1}+X_{2}+X_{3}}{3} follows N_{2}\begin{pmatrix} \mu ,\frac{1}{3}\sum & \end{pmatrix}$

d) The partial correlation coefficients and multiple correlation coefficients lie between −1 and 1.

e) For a renewal function $M_{1},lim \frac{M_{t}}{t}=\frac{1}{\mu }$

04 Dec 2023

Answer :

Word Count : 375

a) True.

A transition matrix $P$ of a Markov Chain represents the probabilities of transitioning from one state to another. The statement is true because the rows of the matrix $P^n$ (where $n$ is the number of steps) represent the probabilities of transitioning from any state to all other states after $n$ steps. As $n$ approaches infinity, the Markov Chain reaches a steady-state, and the transition probabilities stabilize. In this steady-state, all rows of $P^n$ ______ _________ ____ ______ _____ ______ ____ _________ __________.
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