How does the general form relate to the given equation? I don't understand if i just substitute alpha and beta into the general form and that is supposed to give mu=0? Is X2 in the general form the same as Xk? I am just very confused on this notation and how to approach this problem. Thank you in advance!

PROBLEM 1(30pts) Autonomous vehicles must adapt to the spatial environment within which they operate. Adaptation requires that the vehicle shall have a means of characterizing the environment in real time. Consider a drone in a spatially turbulent field. As it flies through that field, the spatial turbulence is transformed into temporal turbulence that is proportional to the speed at which the drone is moving. When flying in a given direction, the classic model for characterizing the turbulence along that direction, beginning at a time index * =0 is: Y = QY_ +U, ; #21 with initial condition Y - N(0, o,), (1a) where (U.) ~ id NO,of =(1-a );) (1b) Letting Y = aY_ gives Y, = QY, + Up. Hence, we see that (la) corresponds to the model Y, = ax, + U , where X, =Y. The only difference is that in the general model we have Y, =\\", + +U . In that model the parameters that minimize the prediction error are: a =o /6, and B= / -au, . Hence, it should be clear that the model (la) assumes u = =0. From (la) we have (i): Y =al, +U,: (ii): Y, = al + U, =a(al, +U )+U, =a'l, +(QU, +U,):, hence, by induction: (2) (a)(3pts) Use (2) to show that, indeed, /, =0. Show all steps. Solution: (b)(4pts) Using (2) to show that E(Y )= o, is straightforward, but cumbersome. Consider the easier alternative method: (: Show that Var(Y, ) = of . (ii): Show that Var(Y, ) = o . [Then, by induction, it is easy to see that E(Y )= ] To show (i) and (ii) use (la) directly, along with information in (1b). [Hint: U is in the present. It is uncorrelated with any random variable in the past.]