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MATH-225 Final Exam — 20.05.2021 — Solutions N.B. Correct answers without sufficient correct mathematical explanations will not get full credit. Q 1: Let A be an n × n matrix, λ1 an eigenvalue of A, and let In denote the identity matrix of size n × n. Recall that the multiplicity of λ1 is the largest integer k such that (λ − λ1) k is a factor of the characteristic polynomial |λIn − A|. (a) (5 pts) Show by an example that the dimension of Null(λ1In − A) can be different f...

MATH-225 Final Exam — Fall 2021 — Solutions Q 1. Let A be a 3 × 3-matrix, B a 3 × 2-matrix, and z a nonzero 3 × 1-matrix. Assume that: (i) B 1 0 0 0 6= B. (ii) B 0 0 0 1 6= B. (iii) B −2 0 0 1 = AB. (iv) det(A) = 2. (v) z is in Null(B T ). (vi) A = AT . Then: (a) (4 points) Compute det(A − 2I). (b) (4 points) Compute rank(B). (c) (4 points) Compute BTAz. (d) (4 points) Compute Az in terms of z. (e) (4 points) If B =   1 1 1 1 2 −1   then com...

Bilkent University Fall Math255 Probability and Statistics Midterm 2 Solutions Problem 1. [6 pts] Let (X, Y ) be uniformly distributed over the triangular region with corners at (0, 0), (1, 0), and (0, 1) in the x − y plane, i.e., fX,Y (x, y) = ( 2, x ≥ 0, y ≥ 0, x + y ≤ 1, 0, otherwise. (a) (3 pts) Compute the probability P 2X2 > Y
. (b) (3 pts) Compute P Z ≤ 3/4
where Z is defined as Z = max(X, Y ). (a) The probability in question is given by 2 (the density...

Bilkent University Fall Math255 Probability and Statistics Midterm 1 Solutions Problem 1. [6 pts] There are three coins each with possible outcomes heads (H) and tails (T). Coin 1 is a fair coin with equally likely outcomes. Coins 2 and 3 are bent coins, having probability of heads 5/6 and 1/6, respectively. Two of the three coins are picked at random and each flipped once. Let A be the event that coin 1 is one of the two coins picked. Let B be the event that the outcomes on the flipped ...

Bilkent University Fall 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 7 Jan 2021 • Write your name, student number, and Math 255 section number on all your answer sheets. Number the sheets. • The exam consists of four problems. 50 points. • This is a closed book exam. One A4-size two-sided page of notes is allowed. No calculators. • You may receive no credit on correct answers if not fully justified. Simplify sums and integrals where possible to receive...

Bilkent University Spring 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 28 May 2021 • Write your name, student number, and Math 255 section number on all your answer sheets. Number the sheets. • The exam consists of three problems. 30 points. • This is a closed book exam. One A4-size two-sided page of notes is allowed. No calculators. • You may receive no credit on correct answers if not fully justified. Simplify sums and integrals where possible to rec...

Bilkent University Spring 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 28 May 2021 Solutions 1. [10 pts] Let X be a Bernoulli(θ) random variable where θ is modeled as a sample of a random variable Θ uniform on [0, 1]. In other words, we use a Bayesian model for the pair (Θ, X) so that fΘ(θ) = ( 1, 0 ≤ θ ≤ 1, 0, otherwise, and pX|Θ(x|θ) = ( θ, x = 1, 1 − θ, x = 0. (a) (5 pts) Compute the Least Mean Squares (LMS) estimate of Θ, namely Θ( ...

P1. (5 points) Let X and Y be two independent drawings from the uniform distribution on [0, a], with a > 0 a given constant. Let Z = X -Y| be the distance between the two points. Find the CDF Fz(z). Show your work by drawing a figure that explains how you calculate Fz(z). 2<0 27 22 Fz(z) = a 92 05259 279 Use this space to show your work for Problem 2 only. S1, if the ith dight drawn is Xi= (o, otheri 7 =) mean of 4th4xyt--+ Xpgon0 10008 (mgon ot X;) 10,00*= 1000 =EEY] ) vav...

That Sugar Film Movie Guide 1.What did the narrator give up eating when he met his girlfriend? 2. What percent of items on the shelves of the grocery store do not contain sugar? 3. How many pounds of sugar do Americans eat a year? 4. How many teaspoons of sugar do Australians eat a day? 5. What is Glucose used for? 6. How must the narrator eat the

Electric Potential & Fields Department of Physics, Case Western Reserve University Cleveland, OH Abstract: In this experiment, we tested the principles of electric potential and fields. The experiment consisted of three different configurations of electrodes, a dipole system, a parallel plate system, and a parallel plate system with a cylinder. We found that in all three setups that our voltage measurements were accurate for the theoretical models. This was determined through our values...