MCB1007 Introduction to Probability and Statistics. First Midterm. Fall Solutions

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İstanbul Kültür University MCB7 Introduction to Probability and Statistics First Midterm Fall 4-5 Solutions Directions You have 9 minutes to complete the eam Please do not leave the eamination room

1 İstanbul Kültür University MCB7 Introduction to Probability and Statistics First Midterm Fall 4-5 Solutions Directions You have 9 minutes to complete the eam Please do not leave the eamination room in the first minutes of the eam There are si questions, of varying credit ( points total) Indicate clearly your final answer to each question You are allowed to use a calculator During the eam, please turn off your cell phone(s) You cannot use the book or your notes You have one page for cheat-sheet notes at the end of the eam papers The answer key to this eam will be posted on Department of Mathematics and Computer Science board after the eam Good luck! Emel Yavuz Duman, PhD M Fatih Uçar, PhD Arzu Yemişçi, PhD Question Question 4 Question Question 5 Question Question 6 TOTAL

2 Queflion Queflion 8+8points (a) How many odd positive integers less than can be formed using the digits,,,, and 4, without repetition? Answer Since we only use the digits,,,, and 4 the number of -digit numbers is: -digit numbers is: 6 -digit numbers is: if the last number is then, if the last number is then 6 thus, +6+(+6)7different numbers can be formed under the given conditions (b) In İstanbul city, all vehicle license plates have letters from the letters of the alphabet followed by 4 one-digit numbers from,,,, 9 How many different license plates are possible for İstanbul if repetition is allowed? Answer There are choices for the first letter For each of these, there are choices for the second letter So, there are 59 possible pairs of letters On the other hand, there are possibilities for each of the first, the second, the third and the fourth digits This means that there are different numbers including the combination Since there is no license plates which ends with the numbers, the total number of different pallets is 59 ( ) points (a) An urn contains si balls numbered through 6 Three balls are randomly drawn from the urn in succession, without replacement What is the probability that the smallest number in this sampling equal to? Answer There are ( ) 6 6! different ways to choose balls from 6 Since is!! the smallest number in this sampling, we need to choose etra numbers from, 4, 5 and 6, which are bigger than two There are ( ) 4 4! 6different ways to do that 4! 4! So the probability that we are seeking for is n N 6 ( ) (b) Find the constant term in the epansion of Answer Since the constant term is the coefficient of the term in the Binomial epansion of the given epression ( ) r r ( )( ) r ( ) r r ( ) 5r ( ) r r r ( ) r ( ) r r r we see that 5r so 5r thus r 4 Therefore, for r 4we have ( ) ( ) 4! ! 6! MCB7 - Int to Prob and Statistics First Midterm

3 Queflion The density function of the continuous random variable X is given by { c( + ), for <<, f(), otherwise 5+points (a) What is the constant c? Answer f()d f()d + ( c( + / )d c + / / f()d + ) ( c + ) f()d f()d c 7 6 c 6 7 (b) Find the distribution function of the random variable X Answer If then F () f(t)dt If << then F () f(t)dt f(t)dt + f(t)dt 6 7 (t + t)dt 6 ( ) t 7 + t/ 6 ( ) / 7 + / If then F () f(t)dt Thus,, ( ) 6 F () 7 + /, <<,, MCB7 - Int to Prob and Statistics First Midterm

4 Queflion 4 points Factories A, B and C produce a tetile product Factory A produces times as many tetile products as Factory B and Factory C Factory A and Factory B produce defective products 5% of the time and Factory C produces defective products % of the time A tetile product is selected at random and it is found to be defective What is the probability it came from Factory C? Answer Let A be event that the tetile product produced in Factory A, B be event that the tetile product produced in Factory B, C be event that the tetile product produced in Factory C, and D be event that the selected tetile product is defective Since Factory A produces times as many tetile products as Factory B and Factory C, we have that P (A) 5, P (B) 5, P (C) 5 Also it is given that P (D A) 5, P (D B) 5 and P (D A) Using Bayes Theorem, we obtain that P (C D) P (C D) P (A D)+P (B D)+P (C D) Queflion points Suppose that a couple will continue having children until have a boy, under the assumption that they have ability to have children and theoretically number of births goes to infinity So, if they have a female child they keep having more children until they have a boy If they have a boy, they stop having children Let X be the number of births Assume that outcomes of births are independent of each other, and boys and girls are equally likely (a) Find the probability distribution of the random variable X? Answer Since boys and girls are equally likely Number of Births to First Boy Point Probability B GB GGB n GGG }{{ G} B n (n ) times thus the probability distribution of the random variable X is f() where,,, (b) Verify that the function in (a) can serve as the probability distribution of X Answer Since f() where,,,, f() is a non-negative function for all in the domain of X On the other hand the series n is a convergent geometric series such that f() n n So, we conclude that the function given in (a) can serve as the probability distribution of X MCB7 - Int to Prob and Statistics 4 First Midterm

5 Queflion points balls are selected at a random from an urn without replacement containing blue and 5 white balls Let the random variable X is the number of blue balls in the first draw and Y is the number of white balls in the second draw (a) Find the joint probability distribution as a table Answer Since P (X,Y )f(, ) ,P(X,Y )f(, ) 8 7 6, P (X,Y )f(, ) ,P(X,Y )f(, ) , thus y g() 5/ / 5/ 6/ 5/ / h(y) / 5/8 (b) Find the conditional distribution of Y given X Answer Conditional distribution of Y given X is w(y ) f(,y) g() f(,y) which can be written as w(y ) { f(, ) for y, f(, ) for y (c) Determine whether or not X and Y are independent Answer X and Y are independent if the equality f(, y) g() h(y) holds true for X, and Y, Let we consider the point where X and Y Since f(, ) 5, g() 5 and h() it is easy to see that f(, ) 5 5 g() h() thus X and Y are dependent MCB7 - Int to Prob and Statistics 5 First Midterm

Question 1. Question 4. Question 2. Question 5. Question 3. Question 6.

Question 1. Question 4. Question 2. Question 5. Question 3. Question 6. İstanbul Kültür University Faculty of Engineering MCB17 Introduction to Probability Statistics Second Midterm Fall 21-21 Number: Name: Department: Section: Directions You have 9 minutes to complete the