Publications by William Jasmine
Data 605 - HW7 - Probability Distributions
library(ggplot2) Question 1 Description: Let \(X_1, X_2,..., X_n\) 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 \(X_i\)’s. Find the distribution of \(Y\). Solution: There are \(k\) possible outcomes for each of the random variables \(X_i\)s...
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Data 605 - Week 7 Discussion
Question comes from “Introduction to Probability” by Charles M. Grinstead. Question 31, Section 5.2 Problem Description: Let \(U\) be a uniformly distributed random variable on \([0, 1]\). What is the probability that the equation: \[ x^2 + 4Ux + 1 = 0 \] has two distinct real roots \(x_1\) and \(x_2\)? Solution: The discriminant \(D\) of a ...
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Data 605 - HW6 - Combinatorics
Question 1 Description: A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2? Solution: There are two possible scenarios to consider: No green (five red) jellybeans are chosen One green and four jellybeans are chosen. We can figure out ...
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Data 605 - Week 6 Discussion
Question comes from “Introduction to Probability” by Charles M. Grinstead. Question 13, Section 4.2 Problem Description: Write a program to allow you to compare the strategies play-the-winner and play-the-best-machine for the two-armed bandit problem of Example 4.24. Have your program determine the initial payoff probabilities for each machin...
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Data 605 - Week 5 Discussion
Question comes from “Introduction to Probability” by Charles M. Grinstead. Question 12, Section 5.1 Problem Description: Prove that the values of the Poisson Distribution given by: \[ \begin{equation} P(X=k) = \frac{\lambda^k}{k!}e^{-\lambda} \end{equation} \] sum together to 1. Solution: To solve this problem, we need simply to take the sum o...
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Data 605 - HW5 - Probability Distributions
Problem 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 dis...
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Data 605 - HW4 - Eigenshoes
Purpose The goal of this document is to take a set of shoe pictures, and build eigenimagery that accounts for at least 80% of the variability seen in said pictures. Essentially, this involves computing the principal components of the combined image data and using them to transform our original images. Gathering Image Data The cell below sets gathe...
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DATA 605 - Week 4 Discussion
Question comes from “A First Course in Linear Algebra” by Robert A. Beezer. Question C27, Page 460 Problem Description: Let \(T:\mathbb{C}^3 \rightarrow \mathbb{C}^3\) be given by \(T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}2x+y+z \\x-y+2z \\x+2y-z \end{bmatrix}\). Find \(\mathcal{K}(T)\). Is \(T\) injective? Solut...
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Data 605 - HW3 - Rank, Eigenvalues, and Eigenvectors
Problem Set 1 Question 1: What is the rank of matrix \(A\)? \[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \end{pmatrix} \] Solution: Upon first inspection, it appears as though none of the rows of the matrix are multiples of each other, meaning that each row is linearly independent and that the rank of ...
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DATA 605 - Week 3 Discussion
Question comes from “A First Course in Linear Algebra” by Robert A. Beezer. Question T10, Page 389 Problem Description: A matrix \(A\) is idempotent if \(A^2 = A\). Show that the only possible eigenvalues of an idempotent matrix are \(\lambda = 0, 1\). Then give an example of a matrix that is idempotent and has both of these two values as eige...
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