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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2011 Math 411 — Problem Set 10 Issued: 11.29 Due: training 10.1. Show that if X is exponential(1), then Y = X/λ is exponential(λ). 10.2. Suppose that X and Y have joint density f (x, y) = c(x + y) for 0 < x, y < 1. Find c and P (X + Y < 1/2). 10.3. Suppose X and Y have joint density f (x, y) = 6xy 2 when x, y ≥ 0 and x + y ≤ 1. Are X and Y independent? 10.4. Suppose a point (X, Y ) is chosen at random from the disk x2 + y 2 ≤ 1. Find the marginal density of X and the conditional density of Y given X = x. 10.5. Suppose that 10% of a certain brand of jelly beans are red. Use the normal approximation to estimate the probability that in a bag of 400 jelly beans there are at least 45 red ones. 10.6. A die is rolled repeatedly until the sum of the numbers obtained is larger than 200. What is the probability that you need more that 66 rolls to do this? 10.7. Among 625 randomly chosen Swedish citizens, it was found that 25 had previously been citizens of another country. Find a 95% confidence interval for the true proportion. 10.8. Suppose we take a poll of 2,500 people. What percentage should the leader have for us to be 99% confident that the leader will be the winner? 10.9. In a 60-day period in Ithaca 12 days were rainy. Is this observation consistent with the belief that the true proportion of rainy days in 1/3?