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📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) is a diagonal matrix, what is the smallest value of \(K\) so that the Manhattan distance between the vector \(\begin{bmatrix} 1 \\ 1 \\ ... \\ 1 \end{bmatrix}\) with ones (\(1\)'s) and its reconstruction using the first \(K\) principal components is less than or equal to ?
📗 Answer: .
📗 [4 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Do not count repeated expansion of the same state. Include both the initial and the goal states.
📗 Answer: .
📗 [3 points] There are \(n\) = students in CS540, for simplicity, assume student \(0\) gets grade \(g = 0\), student \(1\) gets grade \(g = 1\), ..., student \(n - 1\) gets grade \(g = n - 1\). The payoff for each student who drop the course is \(0\), the payoff for the students who stay is if the student has the lowest grade among all students who decide to stay in the class, and the otherwise. If each student only uses actions that are rationalizable (i.e. survive the iterated elimination of strictly dominated actions), how many students will stay in the course? If there are multiple correct answers, enter one of them.
📗 Answer: .
📗 [4 points] You will receive 4 points for this question and you can choose to donate x points (a number between 0 and 4). Your final grade for this question is the points you keep plus twice the average donation (sum of the donations from everyone in your section divided by the number of people in your section, combining both versions). Enter the points you want to donate (an integer between 0 and 4).
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