version D STATISTICS 3083 FINAL EXAMINATION 1998 APRIL 18 7PM THREE HOURS OPEN BOOK AND NOTES CALCULATORS NOT ALLOWED The three questions are of equal weight Where answers are numbers you needn't complete arithmetic: Such expressions as (1.7/sqrt(1+1/7))+(3.14159/2.71828) suffice.
QUESTION ONE A bowl contains four slips of paper, each marked with one of the numbers 0, 1, 3, 8. At random, two (different) slips are drawn without replacement. Call the number on the first slip X, that on the second slip, Y. Since the X and Y are drawn without replacement, X cannot equal Y.
(1)  Complete this table X= 0  1  3  8  SUM of the joint distri +++++ bution of X and Y Y=0  0     1/4 +++++ Answer: 1      1/4 All off diagonal cells +++++ are 1/12. All diagonal 3      1/4 cells are 0. +++++ 8     0  1/4 +++++ SUM  1/4  1/4  1/4  1/4  1 +++++ 

(2)  Are X and Y independent  NO: (1/4)(1/4) is not 0 
(3)  E(X) = 3. Variance(X) = 38/4.
What is the variance of Y?  The same as Var(X) 
(4)  Evaluate Prob(X+Y=1)  1/12+1/12 = 1/6 
(5)  Evaluate Prob(Y=0X+Y=1)  (1/12)/(1/6) = 1/2 
(6)  Evaluate Prob(X+Y=1Y=0)  (1/12)/(1/4) = 1/3 
(7) 
Evaluate the expected absolute value, E( XY )

Average . 1 3 8 1 . 2 7 3 2 . 5 8 7 5 . to get 13/3 
QUESTION TWO For each of these situations. Insert a less than, <, equal to = or greater than > sign to obtain Pr{BA} < Pr{B}, Pr{BA} = PR{B}, etc. A = A network server is overloaded at 4:30 PM Pr{BA} > Pr{B} B = The same server is overloaded at 4:40PM the same day A coin is tossed four times. A = The first two tossed are both heads B = The last two tossed are both tails Pr{BA} = Pr{B} A coin is tossed four times A = The first three tosses are all heads B = The last three tosses are all heads Pr{BA} > Pr{B} A = It rains on a randomly chosen day this year B = It snows the following day Pr{BA} < Pr{B} X and Y are independent random decimal digits. A = (X>5) Pr{BA} = Pr{B} B = (Y>5) A bowl contains 100 slips of paper numbered 0 through 99 One is drawn at random, put back, the bowl mixed well, and another drawn at random. A = The first slip is number 5 Pr{BA} < Pr{B} B = The second slip is number 5 X and Y are independent random decimal digits and S=X+Y A = (X>5) Pr{BA} > Pr{B} B = (S>5) N is a random four digit number A = The first two digits of N are the same. B = The last two digits of N are different. Pr{BA} = Pr{B} An airport has two radar sets call them #1 and #2 A = Radar set #1 fails on a randomly chosen day this year B = Set #2 fails on the same day Pr{BA} > Pr{B} A = A randomly chosen U.N.B. student is now writing the Stat 3083 final B = The student`s best friend is writing the same exam. Pr{BA} Pr{B} The statement is not clear. In grading either < or >, but not = was accepted. 
QUESTION THREE Short answer questions: Answer any 5. If you do more, the first 5 will be graded. [Remember, answers like (4+1.4)sqrt(5+1/9)/17 are OK.] From a deck of four cards, one club, one diamond, one heart, and one spade, a two card hand is dealt. List the sample space. CD CH CS DH DS HS Two digit random (decimal) numbers generated in sequence. A success consists of number with a "7" digit in it. Let T be the number of tries needed to get (exactly) 5 successes. State the name of the distribution of T and its parameters. 19 two digit numbers contain "7" Negative binomial r=5, p=0.19 Evaluate the probability that 6 random decimal digits contain exactly 3 zeroes. 20(9/10)^{3}(1/10)^{3} A continuous random variable X has this distribution. f(X) = C for X in [21,25] = 0 elsewhere (i) Evaluate C. C = 1/4 (ii) Evaluate the expectation of X10 2310 = 13 A normal distribution has mean 22 and standard deviation 6. Evaluate X such that 15.87% of the probability lies below X. 226 = 16 A city has 50 streets of which 20 run northsouth and 30 run eastwest. 20 streets are one way streets, 10 of which run northsouth, the others run eastwest. What fraction of north south streets are one way streets. You must show work for this question 1way 2way Total NorthSouth 10* 10 20* The fraction is the ratio EastWest 10 20 30 of starred numbers, Total 20 30 50 10/20 Eight loaded coins are thrown. For five of these the probability of heads is 1/2, for three coins Pr(Head)=1/5. Calculate the variance of the total number of heads. 5(1/4) + 3(4/25) 