Problem 4. Trajectory estimation, Part | _ 2 points possible (graded) Note: For this problem, you may find this summary useful. (This is also available at the bottom of Lecture 15, 12. Multiple parameters; trajectory estimation.) The vertical coordinate ("height") of an object in free fall is described by an equation of the form a? (t) = 90 + 611'. + 62752, where 90 , 91, and 192 are some parameters and t stands for time. At certain times t1, . . . ,tn, we make noisy observations Y1, . . . ,Yn, respectively, of the height of the object. Based on these observations, we would like to estimate the object's vertical trajectory. We consider the special case where there is only one unknown parameter. We assume that 60 (the height of the object at time zero) is a known constant. We also assume that 62 (which is related to the acceleration of the object) is known. We view 61 as the realized value of a continuous random variable 81. The observed height at time t,- is Y, : 60 + 81151 + 622-2 + m, i : 1,... , n, where W models the observation noise. We assume that 81 N N(0, 1), W1, . . . , Wu N N (0, 0'2), and that all these random variables are independent. In this case, finding the MAP estimate of 81 involves the minimization of 1 n 6513+ 02 2 (yi 60 Blti 62232 i:l with respect to 91. 1. Carry out this minimization and choose the correct formula for the MAP estimate, 01, from the options below. O 6? 2L1 ti (1% 90 922?) 2 0' O ,. ELI if (y; 90 9215?) 0'2 + 2:1:1t12 :7: H | l O A 231% (M 90 92753) 0'2 + 221:1 Hgtf 2. The formula for 91 can be used to define the MAP estimatorl (a random variable), as a function of t1, . . . ,tn and the random variables Y1, . . . ,Yn. Identify whether the following statement is true: The MAP estimator Q1 has a normal distribution. Select an option V Smeit You have used 0 of 1 attempt Save Problem 4. Trajectory estimation, Part II 3 points possible (graded) 1. Let 0' = 1 and consider the special case of only two observations (11. = 2). Write down a formula for the mean A 2 squared error E [(91 91) ], as a function of t1 and t2. Enter t_1 for t1 and t_2 for t2. Eul elf] = 2. Consider the "experimental design" problem of choosing when to make measurements. Under the assumptions of the previous part, and under the constraints 0 S t1,t2 S 10, find the values of t1 and 132 that minimize the mean squared error associated with the MAP estimator. *1 oo- [0 ||