MATH 251, Probability and Statistics I, Fall 2001, Monday Oct. 28, Day 25 Before class

Making up exam points:  You can earn back up to 50% of lost points on a problem by writing a new exam problem that tests the same things I was testing for, and doing the problem correctly.  Hand in the new problem(s), the solution(s), and your (old) exam.
For problem 6, the proofs, learn them, and do them orally on the board, with me asking at each stage why you did what you did, why it is "legal."  Please ask me for help beforehand on any of these. Due Wed. Oct. 31, in class. Till 4:00 pm for oral part. ( In deference to the noble sportswomen.).

Diaconis Article?

Continuing the algebra of means and variances.
 X + Y: the means always add .  Variances add only when X and Y independent.
We can combine the rules:  W = a + bX + cY:  mean, variance?  ("affine" transformation--linear + shift)

Pair with someone who was here/not here Friday.  Work these problems.  If you understand, explain: if you don't, ask:
A) X, Y independent.   µX = 3,  µY = 4, sigmaX =1, sigmaY =2   W = 2 - 4X + Y
::Find the mean and standard deviation of W.
B)  X1, X2, X3 are independent; each has mean 5 and standard deviation 2.
::Find the mean and standard deviation of  their sum,  X1 + X2 + X3.
::Find the mean and standard deviation of  their average, Xbar = (X1 + X2 + X3)/3
QUIZ FRIDAY:  Computing mean and s.d., using algebra of means and variances (pp.334, 337)
-- -- -- -- -- -- -- -- --
Sec. 5.1  Counts and Proportions ("Binomial Distribution")

Read Sec. 5.1.  First I will cover pp. 372-379, then mean and s.d. pp. 380, then  the binomial formula pp. 387-9, Memorize the binomial formula and the mean and s.d. for a binomial distribution. Then pp. 381-385 (proportions and normal approx.)  Skip "continuity correction" pp 386-87.
Hand in: 
Another from the algebra of means and variances:
A)  For the situation of example 4.23, pp. 335-6.  They find µZ there.  They find sigmaX in example 4.24.  Find  sigmaY , and then find sigmaZ
Sec. 5.1:
5.1, 2, 3(briefly) for understanding the binomial setting. 

For the following, define carefully what a "success" is, and its probability, and n, the number of observations or trials.  Then the rest is routine. 
5.11 married? Use formula and table C for a; table C for b&c. 
5.12 players graduate? Use formula and table C for a; table C for b. To use the table, redefine "success" to mean "don't get degree" and restate the probabilities. 
5.15 ESP For parts a, b, c, assume the experimenter has 40 cards, ten of each kind.  She shuffles (thoroughly) all 40, and chooses one.  Then puts it back and shuffles again, repeating 20 times. 
In part d, the deck is shuffled only in the beginning, then cards are taken off the top one at a time and looked at. 
 

 Read, discuss 
 

 

 Optional 

Restating probabilities, exchanging role of success and failure (e.g. 5.12) I use a system like this:  For n = 20, X = # who get degree.  Let Y = # who don't get degree.  X+Y = n.  "p" = .8 for X.  "p" = .2 for Y.  Make a table:
        Possible values
X:  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
Y: 20 19 18 17 16 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1  0
Then circle the values in the x-line corresponding to your probability statement; the values in the y-line below these are the ones for the Y variable probability statement with the same probability
(e.g. X <  1 = {0, 1}in the x line, = {20,19}in the y line, = Y > 19.)
Algebra should  work:  P(X <  1) = P(20 - Y <  1) since X = 20 -Y;  = P( 20 <  1 + Y) =  P( 19 <  Y)   But watch the direction of the inequalities. 

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