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rice and MC ATC cost AVC PA Pa 0 Q, Q, 2. QuantityDate Page No. dy / dx, dy : y = [ x , . .. X p dx ap dimension dy / d x p y = ax, + azk , " ap xp the do of P XT dy =ai X a ap dy F [ dy/ dx, -. dy / dx p - a dx' 1xp VX = A = 2 * 1 2 x 2 ( symmetric ) Q = x' AX ( scalar) = [ x, * 2 ] ] X1 7 9 12 192 2 2 x 2 2 * 1 1x2 2 x1 = [X, X , ] [ xeloe d * 2AXIt 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 consider the markov chain x with state space s="{0," ... and transition probabilities ifj="i-1" puj="10" otherwise for i poo="0," poj="for" j> 1. (a) Is this Markov chain irreducible? Determine the period for every state. (b) Is the Markov chain recurrent or transient? Explain. (c) Is the Markov chain positive recurrent? If so, compute the sta- tionary probability distribution. (d) For each state i, what is the expected number of steps to return to state i if the Markov chain X starts at state i? 5. Consider a Markov chain X = {X} with state space S = {0, 1, 2, ...} and transition probability matrix 0 1 0 0 P 0 0 P = O p 0 q 0 0 . . . 0 0 P 0 4 0 Here p > 0, q > 0 and p+q =1. Determine when the chain is positive recurrent and compute its stationary distribution.2. Markov chain transitions P = [P/j] = Al- AI- NI- Al- NI- AI- NI- AI- Al- Let X1 be distributed uniformly over the states {0, 1, 2}. Let (Xill be a Markov chain with transition matrix P; thus, P(Xn+1=j \\Xn= i) = Pu , i, j E {0, 1, 2}. (a) Is the information source stationary? ~ (b) Find the stationary distribution of the Markov chain (c) Find the entropy rate of the Markov chain