Homeworks

homeworks/Makefile

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+
+all: $(subst .tex,.pdf.open,$(wildcard *.tex))
+
+%.pdf: %.tex
+	pdflatex -shell-escape $< >$@
+
+%.open: %
+	open $*
+
+.SECONDARY:

homeworks/homework1.tex

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+\documentclass[10pt]{article}
+\begin{document}
+\title{BioE241 Homework 1}
+\date{}
+\maketitle
+
+
+\begin{enumerate}
+\item Prove that the detailed balance condition for reversibility of a homogeneous continuous-time discrete-state Markov chain implies the existence of a symmetric matrix that is related to the instantaneous rate matrix by a similarity transformation.
+\item Prove that the matrix exponential, as defined by Taylor series, is a valid solution for the transition probabilities of a continuous-time discrete-state Markov chain.
+\item Prove that a continuous-time finite-state Markov chain necessarily has (at least) one equilibrium probability distribution over states. Under what conditions does it have more than one such distribution? Give an example.
+\end{enumerate}
+
+\end{document}