Homework 1 1. Prove the following properties of Conditional Independence: 1) Decomposition: X Y ,W | Z X Y | Z 2) Weak union: X Y ,W | Z X Y | Z ,W 3) Contraction: X W | Z , Y & X Y | Z X Y ,W | Z 2. Given a series of 1, 2, ..., n. Let’s do a random permutation of this series. There will be some digits keeping their positions after the random permutation. We denote the number of these digits as k. What is the expected value of k? 3. In an entertainment TV program, you are asked to open one out of three doors and there is a big bonus behind one of the three doors. After you choose the door, the TV host tells you that he knows the location of the bonus and opens a door without the bonus. Then, the host asks you “Would you like to change your choice?” What is your answer under the principle of maximum probability? If the host does not know the location of the bonus, will be your answer changed? Please explain the reasons under the principle of Probability. 4. According to your common sense, it should be equal probability to get the upward or downward faces of a coin if you throw it up arbitrarily. It means that the prior probability for “upward” is 0.5. Now, there is a special coin, after 100 runs of experiments, you get 63 “upward” and 37 “downward”. Then, what is the probability that you will get “upward” for the 101-th experiment? 5. What are clique, maximal clique and upward closure? Please explain these terms using the below figures as the examples. 6. Suppose X 1 , X 2 , , X n be i.i.d. samples from a uniform distribution p x 1, What is the expected value of the largest one E max X1 , X 2 , , X n . 0 x 1. 7. Talk about the applications of Probabilistic Graphical Models in your research fields or why you choose this course.