Empirical Analysis of Stationary Distributions in Markov Chains
Consider an empirical transition matrix obtained from a financial market study, representing the probabilities of transitioning between different market states over time. The matrix is given as: \(P = \begin{pmatrix} 0.7 & 0.2 & 0.1 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.5 \end{pmatrix}\) where the states are defined as: – […]
Part a
Compute the stationary distribution for the given transition matrix \(P\). Show that the calculated probabilities add up to 1 when rounded to three significant figures.
Eigenvalues and Eigenvectors of a 2×2 Matrix
Consider the 2×2 matrix \(A = \begin{pmatrix}2 & 3\\1 & 4\end{pmatrix}\). Answer the following: (a) Calculate the trace and determinant of \(A\). (b) Find the eigenvalues of \(A\) by solving the characteristic equation \(\lambda^2 – (\text{trace})\lambda + (\text{det}) = 0\). (c) For each eigenvalue, determine a corresponding eigenvector of \(A\).