PSTAT W120A, Summer 2016 Assignment 5.A Due Wednesday Aug. 31 Directions: Please write up your solutions and submit them on GauchoSpace by the end of the day on Wednesday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded for organization and neatness. Lesson 17 - Geometric Uniform Random Variables 1. Suppose we choose a point uniformly from an equilateral triangle with sides of length 10 inches. What is the probability that the point will be within 1 inch of the triangle's centroid. 2. A house sits in the center of a 300 m by 300 m square lot. If an anthill is built randomly on the lot according to a uniform distribution over the square, what is the probability that the anthill is less than 100 m from the house? 3. Suppose that we drop a round coin with a diameter of 1 cm onto a gigantic sheet of paper with red lines drawn vertically every 10 cm and blue lines drawn horizontally every 5 cm. (a) What is the probability the coin will land on a red line? (b) Find the probability the coin lands on both a red and blue line. (c) Given the coin has landed on a red line what is the conditional probability it will land on a blue line? Are the two events of landing on a blue line and landing on a red line independent? Lesson 18 - Continuous Joint Distributions 4. For (X, Y ) random pairs with joint density f (x, y) = 4(xy x2 y + xy 2 ), over the unit square {0 x 1, 0 y 1} (a) What is the probability of the set {0 x 0.5, 0.25 y 0.5}? (b) Calculate E(X). 5. Suppose that we are waiting for two events to occur. X = the time until the first event occurs and Y = the time until the other event occurs. We model the probability distribution of the times with the joint density \u0014 \u0015 2x y2 3 x y/2 f (x, y) = e e +e 4 for x > 0 and y > 0. (a) What is the probability that X < 1 but Y > 1? (b) What is the probability that X > Y ? 1 of 1