MCS Introduction to Probability and Statistics FIRST MIDTERM

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1 Department of Mathematics and Computer Sciences 1-11 Spring Semester MCS 4 - Introduction to Probability and Statistics FIRST MIDTERM 1 The following data represents the grades of a group of students: 1,17,,34,44,44,48,49,55,59,6,63,7,81,83,89,9 a Find the mean b Find the median c Find the standard deviation d How many standard deviation is the top grade above the mean? Gören, Bengisu, Tuncay, Nurşah, Bahadır and Merve are sitting in a row. a In how many different ways can they sit if boys and girls alternate? b In how many different ways can they sit if Merve and Bahadır are to sit next to each other but Gören and Nurşah are not to sit next to each other? 3 There are three departments in a Faculty: Math, English and Translation. The Math department has 3 times as many students as the Translation, and times as many students as the English department. 4% of all students in the Math department, 8% of English department and 9% of the Translation department are girls. We choose a student from the Faculty at random. Given that the student is a girl, what is the probability that she s a Math student? 4 We pick 3 cards from a deck randomly. We win if they are all the same color red or black or if each of the numbers are less than or equal to 3. Find the probability of winning. 5 X is a continuous random variable with probability density function fx = { k1 x if x 1 if x < or x > 1 a Find k b Find the P.75 < X.

2 Answers 1 a x = 54 b x = 55 c σ = d a 3! 3! = 7 b 5! 4! = Suppose there are n students in Math Dept. Then, there are n n 11n students in translation departments. the total is 3 6. students in English and Let M mean the student is from Math Dept. and let G mean the student is a girl. P M G =.4 n 11 6 n =.4 11 P G =.4 n +.8 n +.9 n 3 = 11 6 n P M G =.4/11 6.6/11 = 4 11

3 4 P A B = P A + P B P A B = 5 3 = a fx dx = 1 k 1 k k = 3 1 x dx = 1 1 x3 3 1 = 1 k 3 = 1 b P.75 < x = x dx =.5 3 = 1 64

4 Department of Mathematics and Computer Sciences 1-11 Spring Semester MCS 4 - Introduction to Probability and Statistics SECOND MIDTERM 1 Given the joint density function fx = 6 x y < x <, < y < 4 8 elsewhere a Find marginal distribution of X. b Find marginal distribution of Y. c Are they independent? Let X have the density function fx = a Find µ b Find E [ X ] 8 x 3 if < x, if x < 3 The probability that a student is ill on the day of an exam is 1. A student has 8 examinations this month on separate days. What is the probability that the student will be ill a on none of the exams? b on exactly two exams? b on more than two exams? 4 We examine a shipment of laptops for defects by choosing a sample of at random. We reject the shipment if there are or more defective ones in the sample. Assume there are 1 defective ones among the. What is the probability that we will a reject the shipment? b accept the shipment? 5 On average there are 7 snowstorms per year in a certain area. What is the probability that there are a Exactly 7, b More than 7, c Less than 7 snowstorms in a given year? 6 MCS 4 grades have a normal distribution with hopefully µ = 55 and σ = 1. Emre hoca wants to fail only the bottom % of the class. What should the passing grade be?

5 Answers 4 1 gx = 6 x y dy 8 = 6x 8 xy 8 y 4 16 = 3 x 4 hy = 6 x y 8 = 6x 8 x 16 yx 8 = 5 y 4 dx gx hy fx, y No, they are NOT independent. 8 a µ = x x dx 3 = 8 x = 4 b E[x ] = = x 8 x 3 dx 3 a n = : =.6634 b n = : c n = 1 : = = n > : =.58

6 a =.68 b 1.68 = Using the table, we obtain: a p7, 7 p6, 7 =.149 or, alternatively, e =.149 7! b 1 p7, 7 =.413 c p6, 7 = Using the table, we see that %8 corresponds to.84 therefore % is.84. z = x µ σ.84 = x 55 1 x = 44.9

7 Department of Mathematics and Computer Sciences 1-11 Spring Semester MCS 4 - Introduction to Probability and Statistics 1 Given the joint density function fx = FINAL Compute the probability that x 1/, y 1/ 3 x + y x 1, y 1 elsewhere The average number of hurricanes per year is 1. What is the probability that on 3 of the next four years, there are 8 or fewer hurricanes? 3 Suppose that airplane engines operate independently and fail with probability equal to.4. A plane makes safe flight if at least half of its engines run. Determine whether a 4-engine plane or -engine plane has higher probability of successful flight. 4 The average time for a trip is 3 minutes with a standard deviation of 4 minutes. Assume distribution is normal. a Find the probability that a trip takes longer than 4 minutes. b The longest 1% of the trips take at least how many minutes? 5 The lifetime in weeks of a certain type of transistor is known to follow a gamma distribution with mean 1 weeks and standard deviation 5 weeks. What is the probability that the transistor will last at most 5 weeks? 6 A certain cable is manufactured with mean strength of 78.3 kilograms and standard deviation 5.6 kilograms. Find the variance of sample mean if sample size is a 64 b 196

8 Answers 1 P = 1/ 1/ 3 x + y dy dx = = = 3 4 1/ 1/ 3 x y + y3 3 3 x + 1 dx / 48 x x 3 1/ dx = 1 16 Using Poisson Distribution Table, probability for one year: P 8, 1 =.155. For 3 of next 4 years: = engine success rate: = engine success rate: =.88 engine plane has higher probability of success a z = =.5 4 Using normal distribution table, P z >.5 = =.6 b P z > 1.8 = =.13 z = 1.8 x = = 35.1

9 5 µ = 1 = αβ σ = 5 = α β α = β = 5 { 1 fx = 5 xe x/5 if x > if x Using integration by parts, we can compute the probability as: p = 5 fx dx = 1 5 e x/5 5 5x = 1 11 e 1 = a σ x = σ n = =.7 b σ x = σ n = =.4

10 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 1 1 Let X, Y have the joint probability function specified in the table. a Compute P X = Y b Compute P Y > c Compute P X 1, Y 3 Y X Answer:.8,.8,.8 Let X, Y have joint probability function given below. Compute P X + Y 1 fx, y = 6 7 x + y, x 1, y 1 fx, y = elsewhere Answer: 3 14

11 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 1 Let X has the density function fx = Find the expected value of X. 3 x x > elsewhere Answer: 4 The random variable X represents the number of defective items in a lot of 1. It has the probability distribution given in the table. Find the variance of X. x fx Answer:.8131

12 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 3 If the joint density function of X and Y is given by: x + y < x < 1, 1 < y < fx = 7 elsewhere find the expected value of gx, Y = X Y 3 Answer:.1

13 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 4 We consider purchasing a welding machine. An efficient machine is successful at 99% of the jobs. An inefficient machine is successful at 95% of the jobs. We perform 1 welds with each machine and purchase the ones that are successful at 97 or more welds. a What is the probability that we will reject an efficient machine? b What is the probability that we will accept an inefficient machine? Answer:.1837,.578

14 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 5 1 Two chess players A and B will meet in 9 matches. The one to win 5 matches wins the tournament. The probability that A will win in a single match is.7. What is the probability that B wins the tournament in 9 matches? Answer:.48 On average, there are 6 misprints per a report. What is the probability that a given report a has no misprints? b has more than 5 misprints? Answer:.5543

Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: 5.

15 Department of Mathematics and Computer Science MCS 4 - Introduction to Probability and Statistics Name-Surname: ID Number: CLASSWORK 6 MCS 4 grades have a normal distribution with µ = 53 and σ = 15. Emre hoca wants to give CC to middle 3% of the class. What should the limits for CC be? Answer: 47.15, Prepared by: Dr. Emre Sermutlu 11

STAT515, Review Worksheet for Midterm 2 Spring 2019

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