İstanbul Kültür University Faculty of Engineering. MCB1007 Introduction to Probability and Statistics. First Midterm. Fall

Transcription

1 İstanbul Kültü Univesity Faculty of Engineeing MCB007 Intoduction to Pobability and Statistics Fist Midtem Fall Solutions Diections You have 90 minutes to complete the exam. Please do not leave the examination oom in the fist 30 minutes of the exam. Thee ae six questions, of vaying cedit (00 points total). Indicate clealy you final answe to each question. You ae allowed to use a calculato. Duing the exam, please tun off you cell phone(s). You cannot use the book o you notes. You have one page fo cheat-sheet notes at the end of the exam papes. The answe key to this exam will be posted on Depatment of Mathematics and Compute Science boad afte the exam. Good luck! Emel Yavuz Duman, PhD. M. Fatih Uça, PhD. Question. Question 4. Question. Question 5. Question 3. Question 6. TOTAL

2 Queflion. Queflion. 5 points Thee ae n maied couples in a paty. All the paticipants shake each othe s hands only once except his/he patne. What is the total numbe of handshakes at the paty? Answe. When two people shake hands, we can think of them as foming a tempoay handshaking committee. The total numbe of handshakes will be the same as the numbe of ways of foming a committee of people fom n people (Thee ae n people at the paty since the paty consist of n maied couples). As the choices ae not odeed, we ae counting combinations; thus the total numbe of handshakes is ( ) n including patnes ones. Since all the paticipants shake each othe s hands only once except his/he patne, consideing thee ae n patnes in the paty we obtain that the total numbe of handshakes at the paty is ( ) n n n! n(n ) n n n(n ) (n )!!! 5+0points A company decided to choose 6 of its employees by dawing and give them a weekend holiday fo evey weekend duing one yea. (a) What should be the minimum numbe of employees of this company if all holiday goups ae diffeent then each othe? Answe. Let n denote the numbe of employees woking fo the company. Since thee ae 5 weekends in a yea, the inequality ( n 5 should be satisfied. Using the definition of a combination, it is easy to see that n 6. Thus fo n 6then ( 6 < 5, fo n 7then 7< 5, fo n 8then ( ) 8 6 8! 8< 5, 6!! fo n 9then ( ) 9 6 9! 84 5, 6!3! So, the minimum numbe of employees of this company is 9. (b) It is given that the numbe of the employees of this company is equal to the minimum numbe that you find in pat (a). Also we know that two bothes ae woking fo this company. What is the pobability of selecting thei names consecutively in the fist dawing? Answe. Let we define an event A {Bothe s names ae selected consecutively in the fist dawing}. Since the numbe of employees woking fo the company is 9, the pobability of selecting thei names consecutively in the fist dawing is P (A) ( ) ) 4 5!! 9P 6 7! 5!! 4!3! 5 9! 36. 3! MCB007 - Int. to Pob. and Statistics Fist Midtem

3 Queflion 3. ( (a) Find the coefficient of in the expansion of x 4 Answe. Using the Binomial coefficient, we obtain x + 3 x ) points ( x + 3 x ) 7 (x ) / 7 ( ) x /3 x ( 7)/ x /3 ( ) x x + 6. Since x + 6 x 4 + 4, 6 thus we see that 3. So, the coefficient of x 4 is ! 3!4! 35. (b) In a goup of 6 maied couple, 4 people ae selected at andom. What is the pobability that NOT maied couple is selected? Answe. Let we define an event A {only one peson fom a couple is selected}. So, the pobability that not maied couple selected is 0 P (A) ( 6 ) 4 4 ( ) MCB007 - Int. to Pob. and Statistics 3 Fist Midtem

4 Queflion 4. 5 points Show that if events A and B ae independent then events A and B ae independent. Answe. Since A and B ae independent events then we know that A and B ae also independent. So, P (A B )P(A)P(B ). On the othe hand, it is easy to see that the B (A B ) (A B ). Since A B and A B ae mutually exclusive, and A and B ae independent by the assumption, we have It follows that P (B )P[(A B ) (A B )] P (A B )+P(A B ) (by Postulate 3) P (A)P (B )+P(A B ). hence that A and B ae independent. P (A B )P(B ) P (A)P (B ) P (B )[ P (A)] P (B )P (A ) Queflion points A continuous andom vaiable X has the following pobability density function { kx 4, x >, f(x) 0, elsewhee. (a) Find k. Answe. f(x)dx f(x)dx + f(x)dx lim kx 4 dx lim k x 3 c c 3 k 3 lim c (c 3 0 ) k 3 (0 ) k 3 k 3. c c (b) Find the distibution function of the andom vaiable X. Answe. If x then F (x) x f(u)du 0 If x> then F (x) x f(u)du f(u)du + x f(u)du x 3u 4 du 3u 3 3 x 3 F (x) x { 0, x,, x 3 x >. MCB007 - Int. to Pob. and Statistics 4 Fist Midtem

5 Queflion points Suppose that 3 calculatos ae andomly chosen without eplacement fom the following goup of 0 calculatos: 7 new, used (woking) and out of ode (not woking). Let X denotes the numbe of new calculatos chosen and Y denotes the numbe of used calculatos chosen. (a) Find the joint pobability distibution table. Answe. Though X can take on values 0,, and 3, andy cantakeonvalues0 and, when we conside them jointly, X + Y 3. So, not all combinations of (X, Y ) ae possible. Since thee ae ( ) 0 3 diffeent ways to choose 3 out of 0, then ( )( f(0, ) ),f(, 0) )( f(, 0) ) 4,f(, ) )( ) 7,f(, ) )( ),f(3, 0) )( )( ) 3) Theefoe, we obtain the joint pobability distibution P (X x, Y y) f(x, y) fo (X, Y ): y x 0 g(x) 0 / / 7/ 4/ / 4/ / 63/ 3 35/ 35/ h(y) 84/ 36/ (b) Find the conditional distibution of Y given X. Answe. Since the conditional distibution of Y given X is given by w(y ) f(,y) g() f(,y) 63/, then w(0 ) f(, 0) 63/ 4/ f(, ) 4/63, w( ) 63/ 63/ / 63/ /63. (c) Detemine whethe o not X and Y ae independent. Answe. If X and Y ae independent then f(x, y) g(x)h(y) fo all x 0,,, 3 and y 0,. Let we conside (x, y) (0, 0). Since f(0, 0) 0 83 g(0) h(0), we see that X and Y ae dependent. MCB007 - Int. to Pob. and Statistics 5 Fist Midtem