We will use the Bayes filter to track the state xt R of a robot. The prior probability density function over the robot state at time t = 0 is p(x) = e^x for x [0, ) and 0 otherwise. The motion model is xt+1 = xt for > 0. The robot is equipped with a sensor that provides observations according to the observation model zt = vt /xt , where the measurement noise vt is independent of xt and has a probability density function q(v) = e^v for v [0, ) and 0 otherwise. Assume that the robot observes before moving.

What is the nonlinear function h and the associated probability density function ph that describe the observation model?

What is the probability density function of the robot state x0 conditioned on the first measurement z0?

What is the nonlinear function f and the associated probability density function pf that describe the motion model?

What is the probability density function of the robot state x1 (after moving) conditioned on the first measurement z0?