Practice Exam 2 Solutions - Statistics for Engineering | STAT 4705, Exams of Statistics

Download Practice Exam 2 Solutions - Statistics for Engineering | STAT 4705 and more Exams Statistics in PDF only on Docsity! Problem 1 Let X be distributed as a gamma distribution with parameters « and §. The density function for X is: E 4 Z eee F(x) T@pet 8 PF >0 a) Prove that the E(x) = aB Ye oo 1 an! EG) = § x: TO c- (ari)-t on 5 b) Show that the moment generating function of a gamma distribution is M,.(t) = =a @ 1 ~*/ =8: e/ yt =F “ng MB Mylt)s Ele ) eS a a Pa) a x & Av. s - , | : . Pea | ae mnt K PCR t) ! Se c) Use the moment generating function for the gamma distribution to show that the var(X) = af? ' een towels = a S a My (= CI ie) ” Heag is as ¢ 2 e cae ee M74) i ) Niet = 222 i BOD 0 A) = GCA BCI Goo lene oo \teo ed ver(x) = EOC) EK) = 2% san | ; 2 4 ) Problem 2 Let Xy and Xp have the joint distribution f(x1,x2) = { Let ¥, = %, + X2 and Y> = Xp a) Find the joint distribution of 9(31,9’2). Y HX te Ye =>¥%2 x= YY Xa= V2 ee uw iD} $C ye)? 24 Cy-ye) Ye [| \ atc b 40 N= = No (ence 7 2G are b) Find the distribution of g(4’, ). iY ar | 24%,%, OSx,5L05x,8L%,+4,81 0 otherwise atterwir ¢ th Jo Cyan gh)  Name. Date. Exam 2 Problem 1 A chemical reacts in a process with probability 0.35. You run 150 independent trials of this process. Let X be the number of times the chemical reacts. a) Describe the distribution of X. (Give the name of the distribution and the mean and variance) 4 X~ dincmml (0,35) wl meme 62.5 & np N@C = 34.125 &— n-P(1-p) b) What’s the probability 2 the chemical reacts in less than 40 of the processes? tiem) = 2 Gar Maas = OWT c) Give a second method you could have used to find the probability in part b. Use this method to find i the probability the chemical reacts in less than 40 of the processes. Normal ferx where Xe Nos | Vai) i 4200) @ Pl 2 ee Y ans PCBS -2.2254) & oi2q Name. Date. Exam 2 Problem 2 You have to take an oral exam to graduate. You will be asked questions until you miss a certain number of questions. Assume the probability of answering any question wrong is 0.08 and they are independent. a) Let Y be the number of questions asked until you get one wrong. State the distribution of Y. (You only need to give the name of the distribution and the mean) \~ geomedrniz ECY) = <7 > 12857 b) What’s the probability you get asked exactly 10 questions? “| Rl yet) = 692)" Cet) = 0378 c) Let Zbe the number of questions you are asked until you answer four questions incorrectly. State | the name of the distribution of Z, and find the probability that you are asked 20 questions. f~ Negative B inevaveed ke 4 e (2 =26) = ¢ A) 92" Lot)! = , fos Problem 3 An electrical firm manufactures light bulbs that have a life, before burn-out, that is normally distributed with mean equal to 800 hours and a standard deviation of 40 hours. a) Find the probability that a light bulb burns out between 780 and 820 hours. Pltwex $28) = RC 7a 22 ee-80? +) =~ e(s 22-4) = Ry — . Beas” wi +383 Name. Date. Exam 2 b) Find the number of hours a light bulb will last if it lasts longer than 90% of the light bulbs. x) vy ) 8 Ze ey | yee x x - you Scape yo Y= oils c) Find the probability that a light bulb burns out after 845 hours. “lx ts) > Pl2> BE) = 7 2 2s) > i.e =. 1272 Problem 4 The life length of a certain component follows an exponential distribution with mean 8 years. a) What is the probability a component lasts more than five years? ( Kr op (s) PLX es) = AXES) = I- C 4 oN ae ee = i-[i? <*) - Sly = css