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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2011 Math 411 — Problem Set 6 Issued: 10.17 Due: 10.24 6.1. A group of 20 people go out to dinner. 10 go to an Italian restaurant, 6 to a Japanese, and 4 to a French restaurant. The fractions of people satisfied with their meals were 4/5, 2/3, and 1/2 respectively. The next day the person you a talking to was satisfied with what they ate. What is the probability that they went to the Italian restaurant? 6.2. Fill in the rest of the joint distribution so that X and Y are independent. There are two possible answers. X=0 X=1 Y =0 ? 2/9 Y = 1 2/9 ? 6.3. Consider f (x) = c(1 − x2 ) for −1 < x < 1 and 0 otherwise. What value of c should we take to make f a density function? 6.4. Suppose X has density function x−2/3 /21 for 1 < x < 8 and 0 otherwise. Find EX, E(X 2 ), and var(X). 6.5. Suppose X has density function F (x) = x/2 for 0 < x < 2 and 0 otherwise. Find P (X < 1) and P (X > 3/2). 6.6. Suppose X has density function f (x) = 4x3 for 0 < x < 1 and 0 otherwise. Find (a) the distribution function F (x) = P (X ≤ x), (b) P (X < 1/2), and (c) P (1/3 < X < 2/3).