23. Let be a function defined and continuous on the closed interval (a,b). If f has a relative maximum at cand a<c<b, which of the following statements must be true? 1. f'(c) exists. II. If f'(c) exists, then f'(c)= 0. III. If f'(c) exists, then f"(c)<0. (A) II only (B) III only (C) I and II only (D) I and III only (E) II and III only
2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a > 0,6 > 0. The beta distribution has density f(x) = p139–(1 – x)6–1, for 0 < x < 1. If X has the beta distribution, show that E(X) = B(a+1,6)/B(a,b). What is Var(X)?
Let A = {x|XEN and 1<x<50} B = { IR and 1<x<50) C= {xxe Z and x 2 25}| Which of the following statements are true? a ACB e V3 EB b. 17 EA [ {0, 1, 2} CA c. ACC g. ØEB d. -40 EC h {xx € Z and > 625} SC
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42 be the change- 01-1 of-coordinate matrix from V to W. Let A be the defining matrix for f under the basis W diagonalize A. 5. Given a linear map f R3R3 if V...
(5) Let X, i = 1,...,n be iid sample from density fx(x) = f(x) e-/201(x > 0), 4 > 0 V TO (a) Find k. (b) Find E(X). (c) Find Var(X). (d) Find the MLE for 0. (e) Find MOM estimator for A. (f) Find bias for MLE. (g) Find MSE of MLE. (h) Let Y = x, find probability density function of Y. (i) Let Y = X?, find cumulative distribution function of Y. 5
Question 7 14 Let f be a twice differentiable function, and let f(6) = 7, f'(6)=0, and f" (6) = 0. Which statement must be true about the graph of f? (6,7) is a local minimum point (6,7) is a local maximum point (6,7) is a global maximum point There's not enough information to tell. (6,7) is a point of inflection (6,7) is a global minimum point Question 5 14.3 pts Let f be a twice differentiable function. y С...
Problem 1. Given the vector space P the basis B -<1,7,',r'> of P, let U - span[1,2]. V-span c and W -spanr x '] Which of the following statements is true? 1. UV = 0 2. UUV is a vector subspace of P -P 3. U nW - and for any vector subspace P of P UW SPP 4. UUW = P. 5. All except statement 3 is false. Problem 2. Consider the function P, R such that f(1-r) -...
Page افت > of 4 Resistance R 4. A square loop of wire with resistance R and side a moves downward and starts entering the constant magnetic field B, att = tz. (See the figure. The loop was outside the field between t = 0 and t =) The loop completely enters the field region and STOPS moving at t = tz. Use the given parameters to put down short answers below. A xxxxx Vi xxxxx x x x x...
ONLY A) B) D) 4 Let X be a single observation from the density f(x; 0)= Ox® -110, 1)(x), where 0 >0. (a) In testing Ho: 0 <1 versus H 1:8 > 1, find the power function and size of the test given by the following: Reject H , if and only if X > . (6) Find a most powerful size-a test of Ho:8=2 versus H 1:0= 1. (c) For the loss function given by [(do; 2) = f(d1;...