Publications by Souleymane Doumbia, Group Member: Fomba Kassoh
Exercise 5, page 362
Exercise 5, Page 362 To achieve the task described, we’ll write an R program that performs the following steps: Chooses 25 random numbers from the range \([0, 20]\) (inclusive) 1000 times. Computes the sum \(S_{25}\) for each set of 25 numbers. Repeats the experiment 1000 times to generate a distribution of \(S_{25}\). Plots the density of \(S_{...
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Proof of Markov Chain
Exercise 17, Chapter 8, Page 324 This problem asks us to show that if \(X\) is a non-negative random variable and \(a > 0\), then the probability that \(X\) is at least \(a\) is less than or equal to the expected value of \(X\) divided by \(a\). We can proof this statement using Markov’s inequality. Markov’s Inequality Markov’s inequality st...
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Document
Problem 11, Page 303 Each of the 100 lightbulbs has an exponential lifetime with a mean of 1000 hours (\(\mu = 1000\)), implying a rate \(\lambda = \frac{1}{1000} = 0.001\) hour\(^{-1}\). We are interested in finding the expected time until the first lightbulb fails, which involves dealing with the minimum of a set of exponential random variables ...
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Trace / Determinant / Factorization
1. Problem set 1 (1) Showing \(A^TA \neq AA^T\) in General Given \(A\) as an \(m \times n\) matrix, \(A^T\) is an \(n \times m\) matrix. The product \(A^TA\) results in an \(n \times n\) matrix, while \(AA^T\) results in an \(m \times m\) matrix. Proof Using an Example: Let’s take a matrix \(A\): \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pm...
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Imp. Distributions / EX / VARX
Problem 1 Distribution of Y, the Minimum of Xi’s Given \(n\) mutually independent random variables \(X_1, X_2, \ldots, X_n\), each uniformly distributed over the integers from 1 to \(k\), we aim to find the distribution of \(Y = \min(X_1, X_2, \ldots, X_n)\). Each \(X_i\) can take any integer value from 1 to \(k\) with equal probability, meaning...
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Combinatorics / Conditional Prob
Question 1 There are only two possibilities for the number of green jellybeans withdrawn to be less than 2: 0 green jellybeans or 1 green jellybean. First case: If 0 green jellybeans are withdrawn Number of green jellybeans withdrawn: 0 Number of red jellybeans withdrawn: 5 Number of ways to choose 5 red jellybeans from the 7 available: \[ \binom...
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Probability Distributions
1- Bayesian Analysis of MNR HIV-1 Test Given: Prevalence of MNR HIV-1, \(P(\text{Disease})\): 0.1% or 0.001 Sensitivity, \(P(\text{Positive} | \text{Disease})\): 96% Specificity, \(P(\text{Negative} | \text{No Disease})\): 98% Calculating the Probability of Having the Disease Given a Positive Test The probability that an individual who tests posi...
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Discussion Assignment
Problem Statement A student must choose exactly two out of three electives: art, French, and mathematics. The probabilities of choosing these electives are as follows: - Art with probability \(\frac{5}{8}\) - French with probability \(\frac{5}{8}\) - Art and French together with probability \(\frac{1}{4}\) We aim to find: 1. The probability that th...
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Eigenvalues / Eigenvectors
Problem set 1 1. What is the rank of the matrix A? \[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \end{pmatrix} \] Matrix A is a 4x4 matrix, then the rank is 4. 2. Maximum and Minimum Rank of a Matrix For a matrix of size \(m \times n\) with \(m > n\), the maximum possible rank is \(n\), which correspond...
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SDoumbia_Assignment4_Eigenimagery_80%
Reading and preprocessing images # Function to read and preprocess images read_and_preprocess <- function(file_name, dir_path) { file_path <- file.path(dir_path, file_name) img <- readJPEG(file_path) list(R = as.vector(img[,,1]), G = as.vector(img[,,2]), B = as.vector(img[,,3])) } dir_path <- "/Users/souleymanedoumbia/Library/Mobile Docume...
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