Publications by Christian Thieme
DATA 605 Week 4 - Singular Value Decomposition & Matrix Inverses
Problem Set 1: A <- matrix(c(1,-1,2,0,3,4), nrow = 2) # A is a 2 x 3 # Transpose will be a 3 x 2 #-------------------------------------------- X <- A %*% t(A) ## A * Transpose(A) #calculating the eigenvalues of X x_eigen_values <- eigen(X)$values #Calculating the eigenvectors of X x_eigen_vectors <- eigen(X)$vectors #--------------...
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DATA605 - Week 3: Eigenvalues and Eigenvectors
Problem Set 1: What is the rank of the matrix A? To find the rank of matrix A, we need to reduce the matrix into reduced row-echelon form and see how many pivot columns there are. We can get the rank from the pivot variables because every pivot variable is associated with a linearly independent vector in the column space.The number of basis vec...
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Combinatorics, Bayes' Theorem, and Conditional Probability
1. A box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places. We can solve this question using the probability of an event rule: \(P(event)=\frac{outcomes\qu...
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Distributions, Expected Value, and Standard Deviation
1. Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y I had a difficult time with this problem. During my research online I came accross this explanation of the problem that helped me understand a ...
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Sum of Random Variables and Law of Large Numbers
Sum of Random Variables and Law of Large Numbers Christian Thieme 10/14/2020 7.11 A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10) In exercise 7.10 we learn, the density for M is exponential with mean μ/n. This means tha...
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Week 9 Discussion - Central Limit Theorem
Chapter 9 Pg. 338 Exercise 1 We’ll solve the below problems using simulation. We’ll set up a uniform distirbution and simulate 100 tosses of a fair coin. We’ll do this 10K times and then count up how many times we met our criteria out of 10K to get our probability. Let \({ S }_{ 100 }\) be the number of heads that turn up in 100 tosses of a...
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Central Limit Theorem & Generating Functions
1.The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by \({ Y }_{ n }\) on the nth day of the year. Finn observes that the differences \({ X }_{ n }={ Y }_{ n }+1-{ Y}_{ n }\) appear to be independent random variables with a common distribution having mean \(\mu =0\) and variance \({ \sigma }^{ 2 }\)=1/4....
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Predicting House Prices: Regression Techniques
Introduction Loading Libraries and Data Exploration and Preliminary Data Cleaning Response Variable - SalePrice Extent of Nulls A Starting Place Overall Quality (OverallQual) and Other Condition Features Above Ground Living Area (GrLivArea) and Other Size Features Other Basement Variables Garage Cars (GarageCars), Garage Areas (GarageArea) and O...
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Computational Mathematics Final Exam
Final Exam - Computational Mathematics Christian Thieme 11/23/2020 Problem 1 Using R, generate a random variable X that has 10,000 random uniform numbers from 1 to N, where N can be any number of your choosing greater than or equal to 6. Then generate a random variable Y that has 10,000 random normal numbers with a mean of \(\mu =\sigma =(N+1)/2...
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Taylor Series Approximations
Taylor Series Approximations Christian Thieme 11/30/2020 Perform the following Taylor Series Expansions: 1. \(f(x)=\frac { 1 }{ (1-x) }\) \(\frac { 1 }{ (1-x) }\) is equivalent to \({ 1\cdot (1-x) }^{ -1 }\) Taking the first derivative we get: \({ -(1-x) }^{ -2 }\cdot (-1)=\frac { 1 }{ { (1-x) }^{ 2 } }\) The second derivative is: \({ -2(1-x) }...
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