R final exam (A), Université Paris-Dauphine, Jan. 18, 2014

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January 21, 2014

R final exam (A), Université Paris-Dauphine, Jan. 18, 2014

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Xi'an

January 21, 2014
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  1. Universit´ e Paris-Dauphine Ann´ ee 2013-2014 D´ epartement de Math´

    ematique Examen NOISE, sujet A Pr´ eliminaires Cet examen est ` a r´ ealiser sur ordinateur en utilisant le langage R et ` a rendre simultan´ ement sur papier pour les r´ eponses d´ etaill´ ees et sur fichier informatique Examen pour les fonctions R utilis´ ees. Les fichiers informa- tiques seront ` a sauvegarder suivant la proc´ edure ci-dessous et seront pris en compte pour la note finale. Toute duplication de fichiers R fera l’objet d’une poursuite disciplinaire. L’absence de document enregistr´ e donnera lieu ` a une note nulle sans possibilit´ e de contestation. 1. Reporter votre login sur votre feuille pour associer votre nom au compte anonyme. 2. Enregistrez r´ eguli` erement vos fichiers sur l’ordinateur, avec un nom de fichier sans accents ni espace, ni caract` eres sp´ eciaux. 3. Sauvegardez votre script pour chaque exercice dans un nouveau fi- chier au nom caract´ eristique comme exercice.2.R. Utilisez le dossier Examen pour stocker ces fichiers. 4. V´ erifiez que vos fichiers ont bien ´ et´ e enregistr´ es en les rouvrant avant de vous d´ econnecter. N’h´ esitez pas ` a rouvrir votre fichier en dehors de R (par exemple avec la commande cat) afin de v´ erifier qu’il contient bien tout votre code R. 5. En cas de probl` eme ou d’inqui´ etude, contacter un enseignant sans vous d´ econnecter ni sortir de R. Il nous est sinon impossible de r´ ecup´ erer les fichiers de sauvegarde automatique. Aucun document n’est autoris´ e, seuls les documents disponibles sur le compte anonyme sont permis. L’utilisation de tout service de messagerie ou de mail est interdite et, en cas d’utilisation av´ er´ ee, se verra sanctionn´ ee. Les probl` emes sont ind´ ependants, peuvent ˆ etre trait´ es dans n’importe quel ordre. R´ esoudre deux et uniquement deux exercices au choix. Exercise 1 1. Write an R function called dart(n,t,l) that (a) uniformly generates n random points (x, y) inside the square [−1, 1] × [−1, 1] and (b) returns the proportion of those points within the unit circle, x2 +y2 ≤ 1. Use dart to derive an approximation of the constant π by this experiment and plot the evolutions of the approximation when the number n of dots grows from 10 to 105. 2. Buffon’s needle is one of the earliest instances of using simulation to approximate an integral. It uses random throws of needles of length over a wooden floor made of planks of width t ≥ and derives the constant π from the proportion of needles
  2. crossing a plank separation (or line). This model assumes all

    planks are identical, ho- rizontal, and parallel. The theoretical probability that the needle crosses the closest line is 2 /tπ. (a) Implement the following R code : BuffonsNeedle <- function(n=100, sd=2){ #(C.) Allan Roberts, 2013. X <- rnorm(n,5,sd) Y <- rnorm(n,5,sd) Angle <- runif(n,0,2*pi) X2 <- X + cos(Angle) Y2 <- Y + sin(Angle) CrossesLine <- ( floor(Y) != floor(Y2) ) p <- sum(CrossesLine)/n return(list(prob=p,esti=2/p)) } and explain the output of the function. (b) Identify from the R code what the corresponding values of and t are. (c) Explain why (or accept the fact that) this R code is only an approximation to Buffon’s problem. (d) For n = 105, give a confidence interval on π and check whether π belongs to this interval. (e) Repeat the above verification with sd=0.2, then with sd=.002. Exercise 2 We wish to estimate the integral I = 2π 0 x2 cos2(x) sin4(x) dx 1. Find the constant k such that f : x → kx2 I{0<x<2π} is a probability density function. 2. Use the generic inversion method to create an R function which outputs n realiza- tions from the density f. 3. Create an Accept-Reject algorithm to simulate from the density g such that g(x) ∝ x2 cos2(x) sin4(x) I{0<x<2π} . 4. What is the estimated acceptation probability of your algorithm ? Deduce an esti- mate of I. Exercise 3 Given the integral I = 3 −3 ex−x2 2 dx
  3. 1. Give the exact value of I using the R

    integrate function (up to a small precision). 2. Propose a Monte-Carlo method using an n-sample from a normal distribution. Give the corresponding 95% confidence interval for I with n = 103. Can we take advantage of the fact that the normal distribution is symmetric to improve our estimator ? 3. Propose an alternative method based on an n-sample from a uniform distribution. Give the corresponding 95% confidence interval for I with n equal to the previous exercise. 4. Establish which method is better by graphical (plotting the evolution of both esti- mates as n increase) and numerical means. Exercise 4 Let us consider a random variable X, whose probability density is given by : f(x) = 3 2 √ xe−x3/2 1 R+ (x) 1. Show that Y = X3/2 is distributed according to the exponential distribution with rate parameter 1, then write a function rsqrtexp(n) that generates n realizations of X. Check graphically that your algorithm is correct. 2. Compute a Monte-Carlo estimate of the expected value and variance of X, together with a 95% confidence interval on the expected value. Illustrate the convergence of the Monte-Carlo estimate to the true expected value (Γ(5/3)). 3. Compute the analytical expression of the cumulative distribution function FX(x). Compute a Monte-Carlo estimate of FX(x) and the associated 95% confidence in- terval, for x ∈ (0.5, 1, 1.5), then compare the estimates to the true values. 4. Compute the analytical expression of the p-th quantile of the law of X. Give the approximate values of the 30%, 60% and 90% quantiles of the law of X, and compare these approximations to the true values.