Reference no: EM133774352
Problem 1: You have a fair coin, a biased coin (probability of head = 0.7) and a fake coin (double head P(head)=1). You choose one of these coins at random and now toss it 5 times. The outcome of each toss is independent of the others.
- What is the probability of Head in the first toss?
- If X is the number of Heads, what is the expected value of X?
- If the first toss is head, what is the expected value of X?
- If in total you get 4 heads in the 5 tosses, what is the probability that you had selected the fair coin?
Problem 2: Let's consider the time as a discrete variable with an increment of 1 minute. You arrive at a bus stop at 10 AM, knowing that the bus will arrive at some time uniformly distributed in the next 30 minutes.
- What is the probability that you will have to wait longer than 10 minutes?
- If, you have waited for 10 minutes and the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?
- Solve part a and b with the assumption that the time of arrival of a bus is a geometric random variable with the expected value of 3.
Probelm 3: A joint PMF is provided as follow:
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X=1
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X=2
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X=3
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Y=1
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0.1
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0.1
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0.2
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Y=2
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0.25
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0.05
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0.05
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Y=3
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0.05
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0.1
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0.1
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Find the marginal PMF of X and Y.
Are X and Y independent?
Find the PMF of Y2 conditioned on X>1
Find E[Y2 |X>1]
Probelm 4. You are designing a strategy for a robot pursing an object. The initial distance between the robot and the object is a random variable D with a known PMF provided in Table 1.
Table 1: PMF of the initial distance between the robot and the object
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d=10
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d=11
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d=12
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d=13
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d=14
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d=15
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P(D=d) 0.1
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0.2
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0.3
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0.2
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0.1
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0.1
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Let assume that all the movements are in one axis. At each time steps (1 unit of time) the object may randomly move to the left, to the right or stop. Let assume that the probability of the object moving to the left and right is the same and equal to 0.4.
The robot will follow the object based on the following strategies (Note that at each time step, the robot will only have one movement).
Strategy 1: At each unit of time, the robot will move 1 unit toward the object regardless of the object movement.
Strategy 2: At each unit of time, the robot will move 1 unit toward the object if the object moves to either left or right, and the robot will stop if the object stops.
Strategy 3*: At each unit of time, the robot will move 1 unit toward the object (with the probability of 0.7) or will stop (with the probability of 0.3) regardless of the object movement.
1. Analytically find the expected value of time for the robot to catch the object using Strategy 1 and 2. Your solution will be a recursive formula.
2. Write a MATLAB code to numerically calculate the above expected values.
Your team need to make one single submissions of the MATLAB code and a PDF document containing the analytical solution and the results.