Estadística I Exercises Chapter 4 Academic year 2015/16 1. An urn contains 15 balls numbered from 2 to 16. One ball is drawn at random and its number is reported. (a) Define the following events by listing corresponding basic outcomes: A The number is even C The number is a prime number B The number is odd D The number is a prime number less than 9 (b) What is the relationship between A and B? Between C and D? (c) Define A B and A B. 2. These are four basic results for two consecutive days of the stockmarket index: O 1 : the index increases both days. O 2 : the index increases the first day, but does not the second day. O 3 : the index does not increase the first day, but it does the second day. O 4 : the index does not increase both days. It is evident that both results must occur, but none of them could occur at the same time. The sample space can be represented as: Ω = [O 1, O 2, O 3, O 4 ]. (a) Considering the following events: A: The index increases the first day B: The index increases the second day Found the intersection, the union and the complement of A and B. (b) Assuming that the four basic results are equally probable. In this case, which is the probability that the market increases at least one of the days? 3. The web page of a retail store receives 1.000 visits per day. Given the past experience, it was found that 1.000 visits gave as result 500 big sales of at least 500 euros and 100 small sales of at most 500 euros. Assuming that the all the visits have the same probability of giving as a result a sale: (a) Which is the probability that a visit gave as a result a big sale? (b) Which is the probability that a visit gave as a result a small sale? (c) Which is the probability that a visit gave as a result a sale? 4. A mobile phone company concluded that 75% of the clients demand the short message service, 80% require a photo app and 65% demand both services. (a) Which is the probability that a client demand at least one of the services? (b) Which is the probability that a client that demand the short message service also require the photo app and that a client that require the photo app also demand the short message service? 5. There are 85 topics covered by an exam. Three topics are selected at random. If a student studied 35 of the 85 topics, what is the probability that she/he studied at least one of the selected topics? 6. In a survey for a TV channel, 2500 people were asked whether they watched a political debate shown by the channel, and whether they watched a movie shown by the channel. Both programs were aired during different times of day. 2100 of the sampled people watched the movie, 1500 people watched the political debate, and 350 people did not see any of the two. If we choose one of the surveyed people at random: 1
(a) What is the probability that he/she watched both the movie and the debate? (b) What is the probability that he/she watched the movie given that he/she watched the debate? (c) What is the probability that he/she watched the debate given that he/she watched the movie? 7. We have two urns: the first one contains 3 red balls, 3 white and 4 black; the second urn contains 4 red balls, 3 white and 1 black. We choose one urn at random and draw one ball. (a) What is the probability that the ball is white? (b) Assuming that the selected ball is white, what is the probability the the ball came from the first urn? 8. There are two bags. In bag A, there are 3 white balls and 7 red. In bag B, there are 6 white balls and 2 red. We draw one ball from bag A and put it in bag B. We shake bag B and select one ball from it. (a) What is the probability that the ball drawn from bag B is white? (b) What is the probability that on both draws we got a white ball? (c) What is the probability that the ball drawn from bag B is white, given that the ball drawn from bag A was white. 9. 38% of Madrid household s incomes are greater than 2000 euros and 37% have income between 1000 and 2000 euros. On the other hand, the percentage of households that have second dwelling is 6.4% for those with income smaller than 1000, 12.57% for households with income between 1000 and 2000 and 23.4% for those with income greater than 2000 euros. (a) Compute the percentage of households that have second dwelling. (b) If a household have second dwelling, which is the probability that the income is greater than 2000 euros? (c) Compute the probability that a household does not have second dwelling and has income greater than 1000 euros. (d) Between the households with incomes greater than 2000 euros, which percentage have second dwelling? 10. We roll 3 dice and report the number of fives we get. (a) Find the probability distribution of the random variable considered. (b) Find its mean and standard deviation. 11. In each of the following situations, say if the variable of interest can be represented by a binomial random variable, and if that s the case, identify the parameters n and p of the distribution: (a) We roll a die 100 times and count the number of 1 s. (b) We deal one card from a deck of 52 cards and check if it is an ace. Without putting the card back, we draw again and check if it is an ace. We repeat this process of drawing without replacement 10 times, and count the number of aces. (c) 2% of oranges that are shipped within Spain are rotten. In a supermarket, oranges come in bags of 10. We select one bag at random and count the number of rotten oranges. (d) In a box there are 2 red balls, 3 are white and 2 are green. We draw one ball at random, write down its colour and put it back in the box. We repeat this scheme 10 times and count the number of white balls. (e) 3% of all items produced by a factory are defective. The items come in 20-packs. We count the number of defective items in a randomly selected pack. (f) In a box there are 2 red balls, 3 are white and 2 are green. We select one ball at random, report its color and return it to the box. We repeat this process 10 times and count the number of balls of each colour. 2
12. We know that 65% of the students of a certain high school will continue their education at a university level. If we randomly select eight students, calculate the probability that: (a) At least one of them will go to university. (b) More than six of them will go to university. How many of them do you expect to go to university? Give-or-take? (Hint: Calculate the expectation and standard deviation of the binomial random variable.) 13. The daily probability of winning a lottery (Monday to Friday) is 0.1. We buy a lottery ticket on five days of a week and we wish to know what is the probability of winning 0, 1, 2, 3, 4, 5 times. (a) Tabulate the six probabilities. What kind of distribution do we have here? (b) Calculate the mean and standard deviation of the distribution from the previous part. 14. The following figure shows a density function of a continuous rv X. (a) Calculate the probability that X is less than 1 using the graph. (b) Calculate the probability that X is greater than 0.5 and less than 3/2 using the graph. (c) Find the mean of X. (d) Find the variance of X. 15. An economist estimated that the sales and costs of some products associated with an index I are given by: If I is a rv (call it X) with the density function Costs: C = I+5 25 I 7, Sales: V = 4. f X (x) = (a) Find the distribution function of I. { x 108, if 3 x 15 0, otherwise. (b) Find the mean and SDs of the costs, sales and earnings. (c) Calculate the probability that that the earnings are negative. 16. Find the following probabilities, where Z follows a standard normal distribution, N(0, 1): (a) P (Z > 0.2). (b) P (Z > 1.27). (c) P ( 0.52 < Z < 1.03). 17. The cholesterol level in a healthy adult follows a N(µ = 192, σ = 12) distribution. Compute the probability that a healthy adult has a cholesterol level: 3
(a) Above 200. (b) Between 180 and 220. 18. 7% of trousers manufactured by a certain company are defective. The trousers come in 80-packs. What is the probability that a randomly selected pack contains more than 10 defective trousers? 19. On a multiple-choice test there are 100 questions with two possible answers: true or false. If a student randomly guesses (all) the answers, what is the probability that he/she will get more than 60 questions right? 20. A rv X has the following density Calculate: (a) distribution function of X, f(x) = (b) the probabilities P (1 < X < 2) and P (X < 1), (c) the expectation and variance of X, { (1 + x 2 )/12, if x (0, 3), 0, if x / (0, 3). (d) the probability P ( X E[X] 1) and the bound you would obtain from the Chebychev s rule. 21. (May 2014 Exam) In a certain company, each visit of the technical service to fix a breakdown in the computer system has a cost of 350 euros, plus a fixed monthly fee of 175 euros regardless of the usage of the service. The monthly average number of breakdowns is 9.5 with a standard deviation of 2. a) Obtain the expectation and variance of the monthly cost of reparations (including the monthly fee). b) By using Chebyshev s inequality, provide a bound for the probability that in a given month the reparations cost is lower than or equal to 2000 euros or greater than or equal to 5000 euros. c) If we now assume that the monthly cost of reparations is continuous uniformly distributed with the expectation and variance obtained in a), compute again the probability of the previous part. d) How can you explain the differences between the results obtained in parts b) and c)? 22. (May 2014 Exam) The lifetime (in days) of a certain product has the following density function: { k (2 x), si 0 < x < 2, f(x) = 0, otherwise, for a given value of k. a) Obtain the value of the constant k. b) We know that this sort of product can only be sold between 12 and 48 hours after it has been manufactured. Calculate the probability that such a product can be sold. (Assume that one day has 24 hours). c) A certain company is able to manufacture 4 products of this kind per day. What is the probability that all the products manufactured during the last day cannot be sold? d) The company will break if all the products manufactured in at least 320 days of the year cannot be sold. Calculate the probability of bankruptcy of the company. (Assume that one year has 365 days). 23. Consider a uniformly distributed rv X with density: { 1/4, x = 1, 2, 3, 4, P (X = x) = 0, otherwise. Let X 1,..., X n be i.i.d. with the same distribution that X, and consider another rv Y = 1 n X i. n Find P (2.4 < Y < 2.8) when n = 36. i=1 4
24. Given the following statements, proof those which are true or find a counterexamples for those which are false. (a) For any rv X, we have that E[X 2 ] E[X] 2. (b) For any continuous rv X with density function f(x) > 0, x [0, 1], and f(x) = 0, x / [0, 1], we have that E[X] > 1. (c) Given two events A and B from a sample space Ω, if A and B are mutually exclusive then they are independent. 25. Let X be a random variable with expectation E(X) = 2 and variance V ar(x) = 1. Then: (a) E(X 2 ) = 5. (b) E(X 2 ) = 4. (c) E(X 2 ) = 3. (d) E(X 2 ) = 4. 26. The following figure contains the cumulative distribution function of a continuous random variable X. the probability that X is greater than 2 is: (a) 0.25. (b) 0.5. (c) less than 0.25. (d) more than 0.5. 27. The following figure contains a representation of a certain function. Which of the following is true: (a) None is true. (b) It is the probability density function of a continuous random variable. 5
(c) It is the cumulative distribution function of a continuous random variable. (d) It is the cumulative distribution function of a discrete random variable. 28. Consider a random variable whose probability density function is equal to a positive constant in the [0, 2) interval and null elsewhere. Which of the following is true: (a) The positive constant is equal to 2/3. (b) It is not possible to deduce the value of the constant. (c) Since the probability density function is not continuous, the cumulative distribution function is not continuous. (d) It is a random variable whose cumulative distribution function is equal to x/2 if x [0, 2). 29. Let (X, Y ) be a random variable that follows a bivariate Gaussian distribution with parameters µ = ( 1, 2) and covariance matrix: ( ) 2 1 Σ = 1 3 (a) Obtain the correlation coefficient between X and Y. How is the linear relationship between both variables? (b) Obtain the generalized variance. (c) Obtain the marginal distributions of X and Y. (d) Are X and Y independent? (e) Obtain the distributions of Y X = 2 and X Y = 1. 30. Let (X, Y ) be a random variable that follows a bivariate Gaussian distribution with parameters µ = (0, 3) and covariance matrix: ( ) 8 4 Σ = 4 12 (a) Obtain the correlation coefficient between X and Y. How is the linear relationship between both variables? (b) Obtain the generalized variance. (c) Obtain the marginal distributions of X and Y. (d) Are X and Y independent? (e) Obtain the distributions of Y X = 4 and X Y = 8. 6