Publications by Souleymane Doumbia, Group Member: Fomba Kassoh
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Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if (a) he bets 1 dollar each time (timid s...
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Probability that the grandson of a man from Harvard went to Harvard # Transition matrix T <- matrix(c(1, 0, 0, 0.3, 0.4, 0.3, 0.2, 0.1, 0.7), byrow = TRUE, nrow = 3) # Calculate the 2-step transition matrix T_2 <- T %*% T # Probability that the grandson of a man from Harvard went to Harvard # Harvard is the first state...
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CLT and Generating Function
1. Exercise 11, page 363 We have been given: mu <- 0 sigma <- sqrt(1/4) Y1 <- 100 days <- 365 According to the Central Limit Theorem (CLT), the sum of a large number of independent, identically distributed random variables, each with finite mean and variance, will approximate a normal distribution. For the...
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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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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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