Publications by Shaya Engelman
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Consider the game of tennis when deuce is reached. If a player wins the next point, he has advantage. On the following point, he either wins the game or the game returns to deuce. Assume that for any point, player A has probability .6 of winning the point and player B has probability .4 of winning the point. (a) Set this up as a Markov chain wi...
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Problem A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (Batting average is the ratio of the number of hits to the number of times at bat.) In the first year, he comes to bat 300 times and his batting average is .267. Assume that his at bats can be considered Bernoulli trials with probability ....
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##Chapter 7.2 Question 15 Suppose we want to test a coin for fairness. We flip the coin \(n\) times and record the number of times \(X0\) that the coin turns up tails and the number of times \(X1 = n − X0\) that the coin turns up heads. Now we set \[ Z = \frac{1}{n/2} \sum_{i=0}^{n/2} (X_i - n/2)^2\] Then for a fair coin \(Z\) has approximat...
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A worker for the Department of Fish and Game is assigned the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of them have tags. (a) Without doing any fancy cal...
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3.2 #31
Introduction This document aims to calculate the probability of a safe landing for an airplane based on the functionality of its engines. Problem Statement Each of the four engines on an airplane functions correctly on a given flight with a probability of \(0.99\). The engines function independently of each other. The plane can make a safe land...
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Data 605 Week 5
1. (Bayesian). A new test for multinucleoside-resistant (MNR) human immunodeficiency virus type 1 (HIV-1) variants was recently developed. The test maintains 96% sensitivity, meaning that, for those with the disease, it will correctly report “positive” for 96% of them. The test is also 98% specific, meaning that, for those without the disea...
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Data 605 Week 4 Discussion C26
C26 Verify that the function below is a linear transformation. \[ T: P_2 \rightarrow \mathbb{C}, \quad T(a+bx+cx^2) = \left[ \frac{2a-b}{b+c} \right] \] To verify if a function is a linear transformation, it needs to satisfy two properties: Additivity: \[ T(u + v) = T(u) + T(v) \] for all vectors \(u\) and \(v\) in the domain of \(T\). Homogen...
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Data 605 Assignment 3
ASSIGNMENT 3 Problem Set 1 (1) What is the rank of the matrix A? \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \\ \end{bmatrix} \] To find the rank of the matrix, we can perform Gaussian elimination to transform the matrix into it’s row-echelon form. First we’ll do Row 2 + Row 1: \[\begin{b...
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Data 605 Week 3
T20 Suppose that \(\lambda\) and \(\rho\) are two different eigenvalues of the square matrix \(A\). We want to prove that the intersection of the eigenspaces for these two eigenvalues is trivial, i.e., \(E_A(\lambda) \cap E_A(\rho) = \{0\}\). Proof by contradiction Assume the opposite, that there exists a non-zero vector \(v\) in both \(E_A(\l...
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Problem set 1 (1) Show that \(A^TA\) \(\neq\) \(AA^T\) in general. (Proof and demonstration.) Since matrix multiplication is not commutative, meaning the order in which you multiply matrices results in different products, AB != BA. AAt, multiplied by AtA is just substituting At for B. This is obviously true for non-square matrices due to the r...
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