Publications by Mael Illien
data605week15exercise
Section 8.8 Exercise 19 Use the Taylor series given in Key Idea 8.8.1 to verify the given identity: \[ \frac{d}{dx}(sin \ x) = cos \ x \] Recall the series representation of sin x: \[ sin \ x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \] \[ \frac{d}{dx}(sin \ x) = \frac{d}{dx} \ (x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7...
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data605week12exercise
Linear Regression Presented below is some data from my current job in real estate development and construction. There are obvious variables that impact the cost of a construction project such as its size in SF. This data below represents construction costs of luxury rental apartments in NYC. Rentals are more or less a standardized product so it a...
2017 sym R (827 sym/4 pcs) 3 img
data605week10exercise
Section 11.1 Exercise 13 Write a program to compute \(u^{(n)}\) given u and P. Use this program to compute \(u^{(10)}\) for the Land of Oz example, with u = (0,1,0), and with u = (1/3, 1/3, 1/3). # Setup library("expm") options(digits=4) # Transition matrix A <- matrix(c(0.5,0.25,0.25,0.5,0,0.5,0.25,0.25,0.5), nrow=3, ncol=3, byrow=TRUE) A ## ...
315 sym R (542 sym/9 pcs)
data605week9exercise
Section 9.1 Exercise 3 A true-false examination has 48 questions. June has probability 3/4 of an- swering a question correctly. April just guesses on each question. A passing score is 30 or more correct answers. Compare the probability that June passes the exam with the probability that April passes it. This problem can be model using the binomi...
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data605week8exercise
Section 7.2 Exercise 14 Assume that X1 and X2 are independent random variables, each having am exponential distribution and density with parameter λ. Show that Z = X1 − X2 has density: \[f_Z (z) = (1/2)\lambda e^{\lambda |z|}\] To avoid confusion, let’s rewrite Z = X1 - X2 as Z = X - Y = X + (-Y) Using the convolution formula \(f_Z(z) = \in...
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data605week7exercise
Section 5.1 Exercise 14 On the average, only 1 person in 1000 has a particular rare blood type. Recall the Poisson distribution \[P(X=k) = \frac{\lambda^k}{k!}e^{-\lambda}\] \[\lambda=\frac{1}{1000}=0.001\] (a) Find the probability that, in a city of 10,000 people, no one has this blood type. Calculate the new parameter \(\lambda'\) \[\lambda' ...
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data605week6exercise
Section 3.1 Exercise 6 In arranging people around a circular table, we take into account their seats relative to each other, not the actual position of any one person. Show that n people can be arranged around a circular table in (n − 1)! ways. Intuitively, we know that the first person has only 1 way to sit since all seats and rotations are i...
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data605week5exercise
Section 2.2 Exercise 4 Suppose you throw a dart at a circular target of radius 10 inches. Assuming that you hit the target and that the coordinates of the outcomes are chosen at random, find the probability that the dart falls (a) within 2 inches of the center. \[P(within\ 2\ inches\ of\ center) = \frac{\pi (2)^2}{pi (10)^2} = \frac{4}{100} = 0...
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data605week4exercise
Exercise LT.C40 \[ if \ \ T: C^2 \longrightarrow \ C^2 \ satisfies \ \ T \Big ( \begin{bmatrix} 2 \\ 1 \end{bmatrix} \Big ) = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \ and \ \ T \Big ( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \Big ) = \begin{bmatrix} -1 \\ 2 \end{bmatrix},\ find\ \ T\ \Big ( \begin{bmatrix} 4 \\ 3 \end{bmatrix} \Big ) \] We start by as...
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data605week3exercise
Exercise EE.C24 Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for A \[ A = \begin{bmatrix} 1 & -1 & 1 \\ -1 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix} \] We are looking for the values of \(\lambda\) that makes \(det(\lambda I_n -A) = 0\) true. In order to solve this, we build the following matrix: \[ \lambda I_n -A = \b...
4013 sym R (527 sym/4 pcs)