Knowing how costs change as output changes is essential to: A. planning and controlling. B. controlling and decision making. C. planning, controlling and decision making. D. decision making and planning. (4pts) Question 11 - Knowing how costs change as output changes is essential to: A. planning and controlling. B. controlling and decision making. C. planning, controlling and decision making. D. decision making and planning.

A stationary distribution of an m-state Markov chain is a probability vector q such that = q P, where P is the probability transition matrix. A Markov chain can have more than one stationary distribution. Identify all the stationary distributions that you can, for the 3-state Markov chain with transition probability matrix O O P Owl Does this Markov chain have a steady-state probability distribution ? 15 pointsBivariate Distributions What is the bivariate probability density function (pdf) and its properties? Marginal, joint, and conditional distributions Expected values and variance calculation in bivariate distributions Marginal, joint, and conditional expectations CovarianceProblem 7.4 (10 points) A Markov chain X0,X1,X2,. . . with state space S = {1,21 3,4} has the following transition graph: (3) Provide the transition matrix for the Markov chain. (b) Determine all recurrent and all transient states. For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is 0 0.5 0.5 0.5 0 0.5 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0 0.7 0 1 0 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.6 0 0.4 1 0 0 regular? (Yes or No) (4) Is the Markov chain whose transition matrix whose transition matrix is 1 0 0.4 regular? (Yes or No) (5) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0.2 0.5 0 1 0