stat 333 - (an Stat 333 Term Test#1 L7 June 7 2006...

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Unformatted text preview: (an Stat 333 Term Test #1 L7 June 7, 2006. 4:30—5:30 pm. NamezM Instructor: C.D. Cutler IDs!— UWuseridzg Instructions 0 Please be sure to enter your NAME, student ID. number, and UWuserid at the top of this page. 0 The test has 8 pages. Please check that you have all the pages. 0 Ordinary scientific calculators may be used. No other aids are permitted. 0 Part A of the test consists of 10 multiple choice questions of equal value. For each question select the one BEST answer. When you complete this part, please enter the LETTERS corresponding to your answers in the blanks provided at the end of the section. Incorrect, unclear, or multiple letters score zero. (You may do rough work in the gaps on the pages or on the backs of pages but rough work will not be marked and cannot be used later to claim credit or partial credit for questions answered incorrectly.) 9 Part B consists of 2 long-answer problems. Answer these in the space provided. And please write legibly as marks will not be given for answers that I cannot read. Marking Scheme: Part A: 10x2=20 PartB: 1.3+4+4=11 2.4+4+6=14 Total for Part B = 25 Total: 20 + 25 = 45 Part A Multiple Choice 1. Suppose that 1% of computer chips produced by a company are defective. There is a quick )1 cheap test which correctly identifies 90% of such defective chips. However, the test also misclassifies about 5% of good chips as defective. If a randomly-selected chip is classified as defective by the test, what is the probability that it is actually defective? (a) 0.154 (b) 0.383 (c) 0.754 (d) 0.9 (e) 1 \\ (w? (/m/‘I‘JM 4r WW/X’W / fl‘ : 05/7/6574 if r M) may We) W . (00/ 27612535256) :' 4c; W H m w my) a mm (mm i) 2. A mouse is placed in the centre of a maze. He has been conditioned, through experience with previous mazes, to know that he will be rewarded with cheese if he can find his way to the exit of the maze. Suppose that three paths lead away from the centre of the maze. The first path leads directly to the exit after 6 minutes of running. The second and third paths lead back to the centre of the maze after, respectively, 4 and 2 minutes of running. If X is the number of minutes it takes the mouse to reach the exit, then E(X) is: (599% (b) 14 1 (c) 14 (e) 00 an: pl) H)+£(xthZ+E(WW , b§+ (L[+£(U734 (uflmj; W weréfls”? 9 '4" ’ H 3. Refer to the above mouse problem. Then Var(X ) is: (a) 48 @32 (c) 200 (d) 144 (e) 248 My. my 4304 Men-Z JUN/V: 1W3: g ; ggigflwiwx‘llyi 4‘ WWW)"; :— 43); i q + wwiuflfié 4 a (20+ 31,203) 3 4. Suppose that X and Y are continuous variables having joint p.d.f. f (as, y) = 4x’1y‘3 for y > a: > 1. Then the marginal p.d.f. of X is: (a) 4x'2 for w > 1 00' gar—13 form>1 / W77j/j? £7 c) :1: for a: > 1 )4 ((1) 2:5’1 for y > a: > 1 (e) 4y'3 logy for y > a: > 1 (I ?< (¢ Q‘t \u 1} ‘1 K14: [/c’ 5. Refer to the above question. Find E(Y | X = x) : i E “L L1 2 2' .3 (a) 2:6 for a: > 1 2C \5 j 17L (b) 45c"1 for x > 1 ‘ I r ,. 1 . « C ((3)1 form->1 fl [W9C LHWfl) {741:} (1’ (“fl/ g; (d) 2 for a: > 1 d 2 ‘ (e) 213—2 fory>z>1 - \ - a? ,1 fill” ( \7/ IX) L‘ {(3531) C glob/‘1 2 - its / N f 'K , Z V / 1'1 l : 223 it “7 fl 7d) i 9‘ x U L l 71 0 _l DC > f / 6. Suppose that the random variable X has probability generating function 3 252/9 Gx(s) = 1—s+%s2 for [sl<3/2 Then E(X) is: (a) 1 (b) 3 (c) 3/2 (d) 14/3 @9/2 r— ) \\ c P 1 7. Major insurance claims come into a company according to a Poisson distribution at the rate of about 10 per month. The amount of a claim is exponentially distributed with a mean of $2000. Suppose claim amounts are independent of one another and of the total number of claims made. Then if X is the total amount (in dollars) of claims over a one-year period then Var(X ) is: (a) 96 x 107 (b) 40,020 x 103 (c) 48 x 104 (d) 4 x 104 (e) 480,240 x 103 N L 10% — h \ 8. Let X and Y be two random variables. Then Var(X) can be obtained as: (a) Var(X) = E(X2 I Y) — E(X I Y)2 (b) Var(X) = E(Var(X I Y)) @ Var(X) = E(Var(X | Y)) + Var(E(X | Y)) (d) Var(X) = E(Y)Var(X I Y) + E(X)2Var(Y | X) (e) none of the above 9. A fair coin is tossed 10 times and the number N of heads is counted. A second fair coin is then tossed until N heads are obtained. Let X be the number of tosses of the second coin required to obtain the N heads. (If N = 0 we set X = 0.) Then E(X) is: (a)5 (b) 15 (c) 8.5 @5 (e) 10 gm : HUN/W) Please enter the LETTERS /correspondin/g to yourC answers in the blanks below: _%5.11 Part B Long Answer 1. Suppose X1, X2, X3, Y are independent uniform random variables over the unit interval (0,1). We define indicator variables 11, 12, I3 by setting I 1 if X j < Y 1‘ 0 fifi2Y fl (a) Derive P(Ij = 1). Hint: condition on Y or use any other valid argument. State your arguments clearly. ”1.9,: MW / MM? MAW/MW A y?” (b) Derive P(Ij = 1k: 1) for j aé k. Hint: again condition on Y or use any valid argument. X54)” AM Mk<y :’ MA J/J'J/K Maw/5414447 PKWY) 2. There are 4 boxes of equal size. A sequence of balls is thrown at these boxes; each ball goes into exactly one box and each box is equally-likely to catch the ball. Moreover the ball tosses are independent of one another. Let N = number of tosses required for all 4 boxes to contain at least one ball. (a) Note that N can be written as N = 1+N2 +N3 +N4 where N2, N3, N4 are independent geometric random variables. This follows because on the first toss one of the boxes catches the ball (we call this the first box) so there are 3 empty boxes remaining. On the second toss the ball either falls into the first box again (in which case the number of empty boxes remains the same) or it falls into a new box (we call this the second box) in which case the number of empty boxes decreases by one. We let N2 count the number of tosses (after the first toss) required for a ball to fall into a second box different from the first box. Then, starting our counting all over again, We let N3 be the number of tosses required for a ball to fall into a third box different from both the first and second boxes. N4 is defined similarly. Find the parameters p2, p3, p4 of the geometric variables N2, N3, N4 . A 2 3H jk (b) Use (a) to obtain numerical values for E (N) and Var(N). State explicitly any properties you are using in these calculations (you may use know troperties of the geometric distribution without proving them.) E(Nl:¥((t/\31+N;W): HEWHU WEN/K): ”1%“va , 2 _ Z V3 {2/} continued next page... ’Uf 8 (c) Derive the probability generating function G N(s) of N (include the interval of conver— gence) and expand it to find P (N = 5). (You may use known properties of the geometric distribution.) N ‘ A/ /1/ \1 ”(I WW I I '2 Z ') I .7 , 761’ ”e ’4 N: H m W3 4M / I I / / (My): E(3N):¥(§'WW?W}: QUJHSNWUSWUSN“) ,,2 ...
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