Publications by Joey Bochnik
Week 6 Discussion Post
Page 88 problem 5 There are three different routes connecting city A to city B. How many ways can a round trip be made from A to B and back? How many ways if it is desired to take a different route on the way back? We have 3 different routes connecting the 2 cities A and B. Each trip is the event and on each trip we have 2 choices: 1.) The route we...
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Week 5 Assignment
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 d...
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Week 5 Discussion post
Page 36 # 9 A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French? We have: \(P(Art) =...
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Week 4 HW
Load in needed libraries library(jpeg) library(foreach) library(EBImage) # Set files to be all the shoe images files=list.files(path='jpg/',pattern="\\.jpg", full.names=TRUE) # print the dimenstions to get the height and width print(dim(readJPEG(files[1]))) ## [1] 1200 2500 3 # Set the height, width and scale variables height=1200 width=2500 sca...
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week 4 discussion post
Page 349 Question C25 Define the linear transformation \(T: \mathbb{C}^3 \rightarrow \mathbb{C}^2\) \[ T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\right) = \begin{bmatrix} 2x_1 - x_2 + 5x_3 \\ -4x_1 + 2x_2 - 10x_3 \end{bmatrix} \] Let \(U, V\) be arbitrary vectors in \(\mathbb{C}^3\) \[ u = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bm...
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JBochnik_Assignment3
Problem 1: Find the Rank of 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 will have to reduce the matrix to row-echelon form and the number of pivots will be equal to the rank of the matrix \[ R_2 = R_1+R_2 \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 7...
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week 3 discussion post
Page 306 question C22 Find eigenvalues of the matrix B: \[ B = \begin{bmatrix} 2 &-1 \\ 1 & 1 \end{bmatrix} \] To find the eigenvalues of B we must find where: \[ det( \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} - \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} ) = 0 \] Simplify: \[ det( \begin{bmatrix} \lambda - 2 & 1 \\ ...
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JBochnik_Assignment2_PS1
Show that \(A^T A \neq A A^T\) in general. (Proof and demonstration.) Proof: Let \(A\) be an \(m \times n\) matrix: \[ A= \begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \\ \end{bmatrix} \] \[ A A^T = \begin{bmatrix} a_{11} & a...
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JBochnik_Assignment2_PS2
function to get L and U matrices from LU decomposition of a square matrix A getLU_Decomp <- function(A) { n <- nrow(A) if (n != ncol(A)) { stop("Input matrix must be square.") } # Initialize matrices L and U L <- matrix(0, n, n) diag(L) <- 1; U <- matrix(0, n, n) # Perform LU decomposition for (i in 1:n) { # Set values s...
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Document
Page 278 C26 A <- matrix(c(2,0,3,2, 5,1,2,4, 3,0,1,2, 5,3,2,1), nrow = 4, byrow = TRUE) A ## [,1] [,2] [,3] [,4] ## [1,] 2 0 3 2 ## [2,] 5 1 2 4 ## [3,] 3 0 1 2 ## [4,] 5 3 2 1 Compute the determinant of A by hand detA_by_hand <- A[1, 1] * (A[2, 2] * (A[3...
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