Consider the DBN in Figure 14.13(b). In the chapter, the battery level Battery_t and the battery meter reading BMeter_t are assumed to be integer-valued with a range of 0 to 5. In this exercise, we will look at a more realistic model where they are continuous variables; for simplicity we will assume a range [0, 1]. 

a. Give exact expressions for the sensor model P(BMeter_t | Battery_t), for both beta distributions and truncated normal distributions, where the mode is at the true value and the standard deviation is 0.1. (Details of these distributions are available in many online sources.) Plot the conditional distributions for Battery_t = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0.

b. Describe in detail a suitable distribution for the transition model P(Battery_t+1 | Battery_t , X˙_t), where X˙_t is a two-dimensional velocity vector. You may assume that the battery drains at a small constant rate r plus an amount proportional to the absolute velocity of the robot, with a standard deviation that is also proportional to the absolute velocity. Remember that the battery charge cannot go below zero and cannot increase. 

c. Explain how to integrate a Charging_t variable into the DBN, which is true just when the robot is plugged into the charging station. 

d. What might a reasonable prior P(Battery₀) look like?