Grau d’Administració i Direcciód’Empresesen Anglès
Grau de Comptabilitat i Finances
Grau d’Empresa i Tecnologia
Statistics I
Set of Exercises 3
Topic 3 - Discrete Random Variables
Professors:
Mikel Esnaola
David Moriña
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ejercicios de estadística, Ejercicios de Administración de Empresas
Asignatura: Statistics I, Profesor: , Carrera: Administració i Direcció d'Empreses - Anglès, Universidad: UAB
Tipo: Ejercicios
2016/2017
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Grau d’Administració i Direcciód’Empresesen Anglès Grau de Comptabilitat i Finances Grau d’Empresa i Tecnologia
Statistics I
Set of Exercises 3
Topic 3 - Discrete Random Variables
Professors :
Mikel Esnaola
David Moriña
- Consider a gamble consisting of tossing a fair die. If the score in the die is an even number then the gambler earns, in euros, twice as much as the score in the die. Otherwise, if the score is an odd number, the gambler loses, in euros, the score in the die. Let X the random variable that measures the gambler’s earnings a. What is the range of X? b. Find the probability function of this variable.
- A box contains 50 red balls, 49 black, and one green ball. Consider a gamble consisting of drawing 2 balls from the box without replacement. If the two balls are both red we lose 2 euros, if the two balls are black we lose 3 euros, if there is one red and one black we lose 5 euros, but if one of the balls is green we earn 200 euros. Find the probability distribution of such random variable.
- A student must complete a test with 4 questions. Each question has 2 possible answers. The student, who has not studied for the test, answers all questions at random. Let X count the number of right answers, and Y the number of wrong answers in the first 3 questions. a. Compute the probability distribution of X. b. What is the probability that the student has at least one answer right? c. Compute P ( X = 2, Y = 3). d. Compute P ( X =2, Y =2).
- Consider the“quiniela” gamble (1-X-2). Suppose i) that one bet (one column) costs 1euro, includes 15 matches and you play only one bed, ii) all the money collected is distributed in prizes, iii) you are the only winner of the gamble, iv) in a “normal” week the total money collected is 2 million euros. Under these conditions, a. Compute the mathematical expectation of the “quiniela” gamble. b. Compute the variance of the gamble. c. Study how the mathematical expectation changes is, ceteris paribus , you play i) one hundred different bets, ii) one thousand different bets. d. Does it pay to play this gamble? Why do you think there are thousands of people that play every week?
- Determine the type of distribution and the associated parameters that better suit to each of the following random experiments: a. Number of out-of-order elevators in a building with 4 identical elevators, each working independently with a 3% probability of malfunctioning. b. Number of people infected with a particular disease among 100 people chosen at random if the probability of infection in this population is 5%. c. Number of people infected with a particular disease among 10 people in a family if the probability of infection in this population is 5%. d. Number of items produced until finding the first faulty item, among those produced in a machine for which 3% of the items turn out faulty. e. Number of customers, out of a total of 30 in the cafeteria, that ordered coffee if 65% of the people orders coffee.
a. The probability that the first question that a contestant fails is question number. b. The probability that only fails 1 in a total of 7 questions.
- Consider two random experiments. In the first we toss a die. In the second we draw at random (with replacement) a ball from an urn containing three balls numbered 1 to 3. Consider the random variables X , which corresponds to the number in the ball, and Y , which corresponds to the number obtained when tossing the die. a. Obtain the joint probability function f ( x, y ) of the random vector ( X, Y ) and construct the matrix of probabilities. Verify that Σ f ( xi, yj ) = 1. b. From the matrix of probabilities recover the marginal probability functions of X and Y. c. Discuss whether X and Y are independent. d. Consider a new random variable Z = X · Y. Get the probability distribution of Z and from it calculate E ( Z ). What is another way we could calculate E ( Z )? Recommendedexercisesfromthe book 100 ejercicios resueltos de estadística básica para economía y empresa: 51, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 67, 70, 73, 90, 92, 93, 94, 95