| 申論題 | (Total 100 Points.) There are 7 problems in this exam. Show intermediate steps and formu-
las for partial credit. You must enplain how you compute your results or answers for full credit.
1. (20 points) There an pussengers in this Hight are from cou e are 300 pa pasengers in an intornational fight. We know that 150, 100, and 50 untries A, B, and C, respectively. Fach pasenger may carry COVID-19 virus with a certain probablity independently. These probablities for the pasaangers from countries A, B, and C are 3%, 4%, and 5%, respectively. 【題組】 (a) If 5 passengeru are selected at random from country C pasengers, ind the probability that there is exactly 1 pussenger carrying COVID-19 virws. (5 points) |
| 申論題 | 【題組】(b) If a pasienger is selected at randoun from the fight, find the probability that this pasenger is carrying COVID-19 virus.(5 points) |
| 申論題 | 【題組】(c) If 5 passengerw are selected a at random from the fight, find the probability that there is exactly 1 pasenger carrying COVID-19 virus. (5 points) |
| 申論題 | 【題組】(d) Suppoee that a passenger is selected at random from the fight and is found to carry COVID-19 virw. Find the probability that the passenger is from country C. (5 points) |
| 申論題 |
2. (20 points) Three balis are drawn
Let X be the number of white balls selocted and Y be the nurber of black balls selected.
n without replacement from 12 balls (4 white, 4 black, and 4 red). 【題組】 (a) Find the joint probability distribution of X and Y. (5 points) |
| 申論題 | 【題組】(b) Find the probability P{Y = 1}. (5 points) |
| 申論題 | 【題組】(c) Find the conditional distribution of X, given that Y -1. (5 points) |
| 申論題 | 【題組】 (d) Find the probability P{X +Y≥2}. (5 points) |
| 申論題 | 3. (10 points) If the joint density function of two random variables X and Y is given by f(x，y) = 2 for 0 ＜x≤y ＜ 1. What is the corre. elation coefielent of X and Y? |
| 申論題 | 4. (15 points) A company that is looking at the time to failure of the parts (days) it uses decides to look 【題組】(a) Pleme test whether the variances of the time to failure of the parts produced by different suppliers are equal, (Please clearly state the null, the alternative hypothesis, and the test statistic, and use a= 0.10) (5 pointa) |
| 申論題 | 【題組】(b) The compuny has decided that if the m longer than it is for the current supplier, it will switch suppliers. Should the company switch mean time to failure for the new su uppliers is signifcantly suppliers? (Please clearly state the null, the alternat use a= 0.05) (10 points) ative bypothesis, and the test statistic, and |
| 申論題 | 5. (15 points) Three different machines are being considered for use in man nufacturing rubber seals. The
of five seals from each machine is nsed to d
machines are being compared with respect to tensile strength of the product. A random sample
machine to machine. The company want to know whether or not the mean tensile strength is the
o determine whether the mean tensile strength varies from
sarne for all three machines. 【題組】 (a) Please construct the ANOVA table. (10 points) |
| 申論題 | 【題組】(b) Please state the null and alternative bypothesis, perform the test, and state your conclusion using 0.05 level of significance. (5 pointe) |
| 申論題 | 6. (10 points) A real estate economist collects data on two similar neighborhoods, one bordering & MRT
station, , and one tha borhood s I about 3 kilometers from the MRT station. She records
50 observations as of real estate values, , where house prices are given in ten thousand NT$ (i.e., NT$ 10000), and size is the number of square meter of living ares. Consider the partial printout for a regression analysis of the relationship between prices (in ten
thousands) and two independent var ariables Size and MRT. As a house can only be near the MRT
station or not, the MRT variable is coded as    【題組】 (a) Plot the least squares S prediction equation associated with two different neighborboods, and discuss your findings. (5 points) |
| 申論題 | 【題組】(b) Test the overal utility of the model (Please clearly state the null, the alernative hypothesis, and the test statistic, and use a = 0.05), (5 points) |
| 申論題 | 7. (10 points) Let X1,X2,..., Xn denote a a random sample from a distribution that is N(θ, 1), where the mean θ is unknown wn. Please show that there is no uniformly most powerful (UMP) test of the simple hypothesis H0 : θ= θ0, where lo be a specified value of θ, against the alternative hypothesis H1:θ≠θ0. |
| 申論題 | 1. (15%) Ten colored balls are selected from four kinds of balls: red, green, white, and black. In
how many ways can these ten balls be selected if 【題組】(a) at least one ball is selected for each color? |
| 申論題 | 【題組】(b) no more than four balls are selected for each color? |
| 申論題 | 【題組】(c) odd number of red balls, odd number of green balls, even number of white balls, and even number of black balls are selected? |
| 申論題 | 2. (15%) Define a relation R on Z by x R y if x - y is even. 【題組】(a) Describe each of the four properties: (i) reflexivity, (ii) symmetry, (iii) transitivity, and (iv) antisymmetry. |
| 申論題 | 【題組】(b) Show that R is an equivalence relation. |
| 申論題 | 【題組】(c) Describe the equivalence classes of the relation R. |
| 申論題 | 3. (10%) For the B-tree below, show the new B-tree that would result from inserting 21. |
| 申論題 | 4. Assume array a[ ] contains seven numbers as shown below (a[0] is not used). 【題組】 (a) (10%) Use Heapsort to sort the array and show the array a[ ] after the complete max heap is constructed (i.e., after phase 1 of Heapsort is done). |
| 申論題 | 【題組】(b) (5%) What is the time complexity of Heapsort? |
| 申論題 | 5. (5%) Consider the five keys on the left side of the figure below. Also as shown in the figure, a
hash function is applied to these five keys where two keys are hashed to the same integer. Discuss
how to solve this collision. |
| 申論題 | 6. Consider a 3-gram index below, where 3-gram refers to a string of 3 characters, and the index
refers to a link list whose nodes are the vocabulary containing the 3-gram. For instance, the first
node in the list is “beetroot”, which contains the 3-gram “etr”. Note that in this data structure,
vocabulary terms are lexicographically ordered. 【題組】 (a) (20%) Discuss how to find the intersection nodes of the two lists (the intersection nodes refer to the nodes appear in both of the two lists), and what is the corresponding time complexity. |
| 申論題 | 【題組】(b) (15%) If the nodes in the lists are not lexicographically ordered, discuss how to find the intersection nodes of the two lists, and what is the corresponding time complexity. |
| 申論題 | 7. (5%) Write the code segment that inserts into a circular doubly linked list the node to which x points before the node to which y points. |
