167hw3 - Game Theory Steven Heilman Please provide complete...

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Game Theory Steven Heilman Please provide complete and well-written solutions to the following exercises. Due February 2nd, in the discussion section. Homework 3 Exercise 1. This exercise deals with subsets of the real line. Show that [0 , 1] is closed, but (0 , 1) is not closed. Exercise 2. This exercise deals with subsets of Euclidean space R d where d 1. Show that the intersection of two closed sets is a closed set. Exercise 3. Define f : R d R by f ( x ) := || x || . Show that f is continuous. (Hint: you may need to use the triangle inequality, which says that || x + y || ≤ || x || + || y || , for any x, y R d . Also, recall that || ( x 1 , . . . , x d ) || = ( d i =1 x 2 i ) 1 / 2 .) Exercise 4. Describe in words the set of points ( x 1 , x 2 ) in the plane such that ( x 1 , x 2 ) (3 , 4). Exercise 5. Let Y be a random variable such that: Y = 2 with probability 1 / 3, Y = 3 with probability 1 / 3 and Y = 5 with probability 1 / 3. What is the expected value of Y ? Let Z be a random variable such that: Z = 1 with probability 1 / 2 and Z = 2 with probability 1 / 2. Assume that Z and Y are independent.
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