Let X1, ..., Xn be a random sample from the distribution 1 f(x; 01, 02) e-(2–01)/02 x > 01, - < 01 <0, 02 > 0. 7 02 Find the method of moments estimators (MMEs) of 04 and 02.
Calculus 4
Let f(x,y) = A)-i-j E) i+j 1. Find the gradient vector Vf (1, 1) at the point (x,y) = (1,1). B) - 1 - 1 D)-i-j 10. . Find the largest value of the directional derivative of the function f(x,y) = ry + 2ya at the point (3,y) = (1,2). A) 53 ' B) V58 C) V63 D) 74 E) 85 y + The function (,y) = 2 + y2 + A) (-3,5), saddle point C) (-1,3), maximum...
20. Let u = (2).= (1) and w = (1) . If ou+yv=w, (x, y € R), then x + y = (A) 1/5 (B) 2/5 (C) 1 (D) 4/5 (E) 3/5
1. [1 points Let L S 10,1 and L E P. For strings x, y e 0,1 of the same length, let x田y denote the bitwise XOR of x and y-eg., 1000田0111 = 1111. Let ㈣ denote the length of z. Let L* L' = {x : 3y, y has lxl/2 ones and x89 E L). Show that L* E NP
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) = {pos, li - j| > 1. Let X and S2 be the sample mean and variance, respectively. a. Show that X is a consistent estimator for u. b. Is S2 unbiased for 02? Justify. - c. Show that S2 is asymptotically unbiased for 02.
20. For X let E(X)-0 and sd(x)-2, and for Y let E(Y)--1 and sd(Y)-4. Find: (a) E(X-Y) and E (X Y). (b) Var(X- Y) and Var(X+ Y) if X and Y are independent. (c) EGX+ 흘 Y) and Var(1X+] Y) İf X and Y are independent. (d) Repeat (b) if, instead of independence, Cov(X, Y)- 1. soY is VarY larger
Let Z ~ N(0,1) and let Y = Z2. Find the distribution of Y. Hint: Use moment generating function. Let X ~ N(j = 1, 02 = 4). If Y = 0.5*, find E(Y?). Hint: Use moment generating function.
(I point) Let F=21+(z + y) j + (z _ y + z) k. (1+4t). y = 4 + 2t, z = _ (1+t). Let the line l be x =- (a) Find a point P-(zo, 30, zo) where F is parallel to 1. Find a point Q (which F and I are perpendicular. Q= and l are perpendicular Give an equation for the set of all points at which F and l are perpendicular. equation:
(I point) Let F=21+(z...
5. Consider the language L = {1'0/1k e {0,1}* |i >01) >0 Ak = i*j}; to show that Lis! not a regular language using pumping lemma, the correct choice for the word is: a. 10011 x=1- b. 1POP 1P Z=1 Le 1290? 12p* Z=1P OPS 2P Y-101 YEK d. 10P1P y=1" t:P
3. Let Y be a random variable whose probability mass function under Ho and Hi is givern by 1 23 4 5 6 7 f(yHo) .01 01 01 01 01 01 94 fulHi) 06 0504 .03 02 01 79 Use the Neyman-Pearson Lemma to find the most powerful test for Ho versus Hi with Use the Nevmam-Pearson Lemma to find the mst size α-004. Compute the probability of a Type II error for this test.
3. Let Y be a...