BASIC DOCTORAL LEVEL WRITTEN EXAMINATION IN BIOSTATISTICS PART I (9:00 a.m.,22:00 p.m., Friday, August 5, 1994) INSTRUCTIONS: a) This is a closed-book examination. b) The time limit is five hours. c) Answer any four (but only four) of the five questions which follow. d) Put the answers to different questions on separate sets of papers. e) Put your code letter, not your name, on each page. f) Return the examination with a signed statement of the honor pledge on a page separate  from your answers.  g) You are required to answer only what is asked in the questions and not all you know about  the topics   186-940811  a) Define a time-dependent Poisson process. Does it have the Markov (process)   property?  b)Let N(t) be the number of domestic violence events occurring in a town in a time  period [0, t], t > 0. Suppose that each occurrence has a constant probability p;(0 < p; < 1) of being reported to the police department, and these reportings are made independently of each other as well as of N(t). Show that if N(t), t > 0, is a time-dependent Poisson process with intensity l;(t), t > 0, then M(t), the number of occurrences reported to the police department in the time interval [0, t], t > 0, is also a time-dependent Poisson process with intensity m;(t) = p;l;(t), t > 0.  c)For a Poisson process with parameter l;, it is given that N(t) = n for t > 0. Show  that the density function of the time of occurrence Zk up to the kth event (k < n) is given by  f(x) = V0 kB0nkK0 t-k xk-1 (1,2x/t)n-k,8 0<8x<8t, 08,8x;2t.8 Find the mean of Zk.  187-940812 (Kupper) If Y is a normally distributed random variable with mean m; and variance s;28, then the random variable X = eY has a lognormal distribution. The lognormal distribution has been used in many important practical applications, one such use being to model the distribution of exposure concentration levels to which workers in occupational settings are exposed.  a) Using the fact that Y [2 N(m;, s;28) and that X = eY, prove that  #E(X) = e(m;+208.85808s;28)  and that  #V(X) = e(28m;+2s;28) (es;28,21).  b)If the lognormal random variable X = eY defined in part (a) represents the exposure  concentration in parts per million (or ppm) to which a worker is exposed, and if E(X) = V(X) = 1, find the exact numerical value of pr(X>1), namely, the probability that a randomly chosen worker will be exposed to an exposure concentration level greater than 1 ppm.  c)To protect the health of workers, it is desirable to be highly confident that a worker  will not be exposed to an exposure concentration greater than C ppm, where C is a known positive constant specified by federal guidelines. Prove rigorously that  #pr(X :2 C) ;2 (1,2a;), 00. Let X18, X28, ..., Xn constitute a random sample of size n from fX(x; h;r). Analogously, for lifetime residents of United States urban areas, let the distribution of Y, the proportion of this same component in a cubic centimeter of blood taken from such an urban resident, be  fY(y; h;u) = h;u yh;u,21, 00. Let Y18, Y28, ..., Ym constitute a random sample of size m from fY(y; h;u).  a)Using all (n + m) available observations, find two statistics that are jointly  sufficient for h;r and h;u.  b)Show that a likelihood ratio test of H08: h;r = h;u (= h;, say) versus H18: h;r W2 h;u  can be based on the test statistic  T = 0nȶi=1 ln(Xi) / [ 0nȶi=1 ln(Xi) + 0mȶi=1 ln(Yi)].  c)Find the exact distribution of the test statistic T under H08: h;r = h;u (= h;, say), and  then use this result to construct a likelihood ratio test of H08: h;r = h;u (= h;, say) versus H18: h;r W2 h;u with an exact Type I error rate of a; = 0.10 when n = m = 2. 189-940814 (Kupper)#[Note: (-see also 155] Let (Xi, Yi), i = 1, 2, ..., n, be a set of n mutually independent pairs of random variables, where the conditional distribution of Yi given Xi = xi is N(a; + b;xi, s;28).  a)With X~ȫm2 = (X18, X28, ..., Xn) and x~ȫm2 = (x18, x28, ..., xn), derive explicit expressions for  E(b;^8 | X~ = x~) and V(b;^8 | X~ = x~), where  #b;^8 = 0nȶi=1 (Xi,2X,2) Yi 0nȶi=1 (Xi,2 X,2)28 , with X,2 = n-1 0nȶi=1 Xi.  b)If X18, X28, ..., Xn constitute a random sample of size n from a N(m;x, s;28ɬx)  population, derive explicit expressions for E(b;^8) and V(b;^8), the unconditional mean and variance of b;^8.  c)With the same assumptions as in part (b) for X18, X28, ..., Xn, suppose that Zi = Xi  + Ui, where Ui[2N(0, s;28ɬu), where U~ȫm2 = (U18, U28, ..., Un) is a vector of mutually independent random variables, and where the elements of U~ are independent of the elements of X~. If  b;^8ȫ*8 = 0nȶi=1 (Zi,2Z,2) Yi 0nȶi=1 (Zi,2 Z,2)28 , where Z,2 = n-1 0nȶi=1 Zi, derive an explicit expression for E(b;^8ȫ*8), and then comment on the use of b;^8ȫ*8 as an estimator of b;.   190-940815 (Sen) Suppose that there are k(;22) groups, each group generating its own 2 -2 2 table. For the ith group, we have  #Outcome variable Sample present#(-2absent7