Question

You are designing a new robot to give campus tours. But while testing the robot in Aldrich Park, it malfunctions and begins driving randomly along a path. Let X = 0 indicate the robot’s position when the malfunction occurs, positive X values indicate movement to the north, and negative X values indicate movement to the south.

a) Suppose that every minute, the robot travels south 5 meters with probability 1/2, or north 5 meters with probability 1/2. The directions of travel at successive minutes are indepen- dent. After one hour, what is the expected position of the robot? What is the standard deviation of the robot’s position?
b) Suppose instead that the motor controllers on the robot are biased, so that every minute it travels south 5 meters with probability 3/4, or north 5 meters with probability 1/4. The directions of travel at successive minutes are independent. After one hour, what is the expected position of the robot? What is the standard deviation of the robot’s position?
c) You were taking a break when the robot began malfunctioning, and did not observe its movement. Under which motion model, the one from part (a) or part (b), do you think it would be faster to find your lost robot? Justify your answer.

Answer 1

Part a ; Expected position will be 0 (at origin)

and Standard deviation will be 0 .

Part b ; Expected position will be 150 meters South

and Standard deviation will be 12.24.

Part c ;

Using Model motion of part (a) , it will be easier to find the lost robot.

Explanation:

Part a ;

To find out the expected value we have to multiply the probability with the distance so that

x represent the distance/hour while the P(x) represent the multiplication

X=5 meters *60 minutes

X=300 meters / hour

P(x) = 300*(1/2)

P(x)=150

as North represent the positive distance while south represent the negative distance and their probability is also same +150 meters north and -150 meters south

To get expected value we add both the P(x) which result in 0

and for standard daviation taking it square root.

Part b;

Do same as part (a) but there is a change in probability so consider this fact.

Part c ;

By comparing the final answer of part a and part b you will clearly see the wide difference in the distance covered by the robot in both cases. So it is easy to justify that the robot for first motion model will be faster to find.