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:
Alicia Gómez Tello
David Moriña
Xavier Vilà
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Probability Distributions and Expected Values in Economics and Business - Prof. Vilá, Apuntes de Administración de Empresas
A set of exercises on probability distributions and expected values in economics and business. The exercises cover various topics such as discrete random variables, probability functions, mathematical expectations, and variances. Students will learn how to find the probability distribution of a random variable, calculate the expected value and variance, and determine if two random variables are independent.
Tipo: Apuntes
2014/2015
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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 :
Alicia Gómez Tello
David Moriña
Xavier Vilà
- 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).
- Considerthe“quiniela” gamble (1-X-2). Supposei) that one bet (one column) costs 1euro, includes 15 matches andyou 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 2million 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, ceterisparibus , you playi) one hundred different bets, ii) one thousand different bets. d. Compute anew if, besides the changes in item c., the organization distributes in prizes only 50% of the money collected. e. Compute anew if, besides the changes in items c. and d., you are not the only winner and thus must share the prize withi) another winner, ii) other nine winners. f. 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.
a. The probability that a pack is considered defective b. The probability that a box has 2 defective packs.
- The number of accidents that occur in a city is, on average, two per week. a. What is the probability of not having any accident in a given week? b. What is the probability of having exactly 3 accidents in two weeks? c. What is the probability that in a month more than 2 accidents occur?
- In a TV game show the probability that a contestant guesses right a question is 0.60. Find: 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