Bivariate joint

Consider the bivariate joint PDF of X and Y given by (c is some constant) fxy (u, v) = otherwise 1. Find the value of c. 2. Find p(X Sys 3. Find fy(v) the marginal distribution of Y. 4. Compute p(Y |Y = ) 9. Determine whether X and Y are independent.2. (15 pts) Consider a Markov chain { Xn } with state space S = {0, 1, 2} and transition matrix and transition matrix P = O ON/H HN/H O (1) Let the mapping f : S - S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) # f(2). If Yn = f(Xn), then when is { Yn } a Markov chain? Is {Yn } always a Markov chain? In other words, are functions of Markov chains always Markov chains?4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 P= I-T- VVO 0 0 1 Is it possible to choose values for ar and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. (b) The Markov chain is reducible. UNNN (c) The Markov chain has at least one transient state. (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4).A turbojet engine has a turbine that receives combustion gases at 1200 K and 15 bar from the engine combustion chamber. These gases are then fed from the turbine exit to a nozzle, which exhausts to the atmosphere at 0.9 bar. If the combustion gases can be considered air, which is an ideal gas mixture, and both turbine and nozzle are isentropic, what are the temperatures at the turbine and nozzle exits, and the velocity at the nozzle exit? The gases at the turbine exit are at 3 bar and have negligible velocity. Draw a labeled process flow diagram of this process as part of your solution