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It is 0.831999 Bivariate Fit of Happiness By LifeExpectancy Happiness un 40 45 50 55 60 65 70 80 85 LifeExpectancy Bivariate Normal Ellipse P=0950 Bivariate Normal Ellipse P=0990 Bivariate Normal Ellipse P=0.950 Variable Mean Std Dev Correlation Signit. Prob Number LifeExpectancy 67.83646 11.04193 0.631999 <.0001 happiness bivariate normal ellipse p="0.990" variable mean std dev correlation signit. prob number lifeexpectancy can we conclude that being happier causes people to live longer explain. pts explain what is meant by the transition probability matrix of a homogeneous markov chain. marks stationary distribution chain has with entries ouj and . write down an expression for reverse consider following homogo- neous three states i k tpm in order if vector determine whether reversible. let x1 x2 xa be sequence random variables resulting from above xs="j" standard chessboard x grid possible locations. define randomly moving single chess piece on this board. initial location xo sampled uniformly among squares. at time t then chooses xt sampling set legal moves given its current xt. description see: http: wikipedia. org suppose king which move any adjacent irreducible aperiodic b bishop. c knight. state space find proportion does spend long run x5="1." expected additional steps until first will return>