1. Consider the following digraph, where we have a link (xi, xj) for each I s...
1.30 Consider the twelve configurations ci, 1 <i< 12, in Figure 1.38. Draw the digraph D, where V(D) edge of D if it is possible to obtain cy by rotating the configuration c either 90° or 180° clockwise about the midpoint of the checkerboard. {c1, c2,...,C12} and where (ci,c) is a directed D2: Di: Figure 1.37: Digraphs considered twelve configuratione Example 1.3, where we Next, we return to of two coins (one silver, one gold), which were denoted by ci,C2,.....
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3} a. List all numbers of S. b. List all ordered pairs in R. c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)? d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges. e. Give each edge in...
Suppose that the covariates Xj,i for i 1, 2, , n and j 1, 2, , indicator variables for a single categorical variable in the manner covered in the course. Thus, suppose that for each individual i = 1,2,…,n we have that X1.i, X2.i,...,Xd,i this one is equal to the number 1. Let Bk be the (A , . . . , β 1), the minimizer of L (bi , b2, . . . ,勿of eq. (B. = Yn.(k), where...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
1. Given data on (yi, xi) for i = 1, , n, consider the following least square problem for a imple linear regression bo,b We assume the four linear regression model assumptions dicussed in class hold (i) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa tions are called normal equation'. (Hin wo n-dimesional vectors (viand (wi)- are normal-orthogonal ) if Σ-1 ui wi-0. )...
Problem 5.10.10 Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2, …, Xn are iid continuous uniform (0, m) random variables. (Think of a number like one million for the symbol m.) Unfortunately, you don’t know Xi until you open suitcase i. Suppose you can open the suitcases one by one, starting with suitcase n and going down to suitcase 1. After opening suitcase i, you can either accept or reject Xi dollars. If...
Problem 5.10.10 Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2, …, Xn are iid continuous uniform (0, m) random variables. (Think of a number like one million for the symbol m.) Unfortunately, you don’t know Xi until you open suitcase i. Suppose you can open the suitcases one by one, starting with suitcase n and going down to suitcase 1. After opening suitcase i, you can either accept or reject Xi dollars. If you...
a) In lecture we derived the estimate of B in WLS as Derive A.wls when p-1. (It should have a form similar to simple linear regression.) (Hints: Notice that we can write a weighted average as analogues of the sums of squares identities we've used; you should derive these if you need to use them.) . You may need to use weighted b) Assume we have the following data: T1 T2 y2 That is, we have a total of n...
3. Consider the generalized one-dimensional Ising model with link-dependent interactions J. In other words, we have N spins s, in a chain and energy N-1 SiSi+1 We recover a standard Ising model (with B 0 and open boundary condi tions) if we set JiJN-1-J a) Show that the canonical partition function is Hint: You can do this by induction and carefully comparing the states of the N-spin chain to those of the (N - 1)-spin chain.] b) Show that the...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...