MIT OpenCourseWare http://ocw.mit.edu 18.443 Statistics for Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.443 Problem Set 9 1. Rice, §13.8, Problem 18, but replace each number in the table by itself minus 4. So replace 39 by 3 5, 113 by 109, 15 by 11, and 15 0 by 146, so we would now have 3 01 pai rs instead of 317. Test if a mong the 301 controls, t he probability of having taken estrogen is the same as the probability that the matched case patient had taken it. (That’s just a restatement of the problem, not an additional part.) Omit the question in the last sentence. 2. (a) Rice, Sec. 10.9, problem 16, but give formulas for the Q-Q plots. You don’t need to graph them. (b) Let Φ be the N(0, 1) distribution function and G another distribution function. Suppose that the Q-Q plot for G vs. Φ is a straight line y = ax + b. (What are the p ossibilities among a < 0, a = 0, or a > 0?) Show that G is the distribution function of some normal N(µ, σ2) distribution and evaluate µ and σ in terms of a and b. Hint: Part ( a) gives examples. 3. Rice, §10.9, Pro blem 27, but only from days 15n to 15n + 1, n = 0, 1, . . . , 8. Data for this are prov ided in the file “ medflies.txt ” o n the course website. 4. A Pareto distribution for each p > 1 has the density fp(t) = cp/tp for t ≥ 1 and fp(t) = 0 for t < 1. (a) Evaluate the constant cp as a function of p. (b) For the Pareto distribution for each p, evaluate the survival function S(t) = 1 − F (t). (c) Evaluate the hazard function. (d) Answer Problem 14, §10. 9 of Rice. 5. The Cauchy distribution is one having density f (x) = 1/(π(1 + x2)) for all real x. This is the same as the t distribution with 1 degree of freedom. (a) Show that the distribution function is F (x) = a + b arctan x for some constants a and b and evaluate the constants. Hint: the derivative of the distribution function must give the density, so evaluate b from that. Then find a from the properties of distribution functions when x g oes to ±∞. (b) Recall that g(x) ∼ h(x) means g(x)/h(x) → 1, in this case as x → +∞. Show that for the Cauchy distribution, the survival function S( x) satisfies S(x) ∼ 1/ (πx). For the t1−α quantiles with df = 1 given in the t t able, check how accurate t he approximation is for α = 0.05, 0. 025, 0.0 1, and 0.0 05. (c) For any i.i.d. random variables X1, ..., Xn with distribution function F , show that the distribution function of the maximum observation ( i.e. the largest order statistic X(n)) is nF . (d) Show that an approximate median of the distri bution of X(n) for a Cauchy distribution as n becomes large i s cn for a constant c > 0 and evaluate the constant. Hint: for any real number a, � � na 1 − → e −a n as n → ∞ , as can be seen by t aking logs o f both sides and using the derivative of ln at 1.