Stat 3083 final of 1998 April with solutions
                                                            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=0|X+Y=1) (1/12)/(1/6) = 1/2
(6) Evaluate Prob(X+Y=1|Y=0) (1/12)/(1/4) = 1/3
(7) Evaluate the expected absolute value, E( |X-Y| )
(Since all probabilities are the same, 1/12, a simple average of these numbers works)

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{B|A} < Pr{B},  Pr{B|A} = PR{B}, etc.

 A = A network server is overloaded at 4:30 PM            Pr{B|A} > 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{B|A} = 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{B|A} > Pr{B}

 A = It rains on a randomly chosen day this year
 B = It snows the following day                           Pr{B|A} < Pr{B}

 X and Y are independent random decimal digits.
 A = (X>5)                                                Pr{B|A} = 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{B|A} < 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{B|A} > 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{B|A} = 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{B|A} > 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{B|A}   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 X-10                    23-10 = 13

 A normal distribution has mean 22 and standard deviation 6.
 Evaluate X such that 15.87% of the probability lies below X.
                                                         22-6 = 16
 

 A city has 50 streets of which 20 run north-south and 30 run east-west.
 20 streets are one way streets, 10 of which run north-south, the others
 run east-west. What fraction of north south streets are one way streets.
 You must show work for this question
                   1-way 2-way  Total
      North-South   10*   10     20*        The fraction is the ratio
      East-West     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)