MATH 251, Probability and Statistics I, Fall 2007, Mon. Oct. 22 Day 25.after class.

Read Sec. 5.1.  First pp. 334-40, now mean and s.d. pp. 340-1, and the binomial formula pp. 348-50, Memorize the binomial formula and the mean and s.d. for a binomial distribution. Then pp. 341-347 (proportions and normal approx.)  Skip "continuity correction" pp 347-48 (or read optionally).
Hand in: 
5.9a  Random digits as binomial  (Note, P("at least one") = 1 - P("none")--check it out on a list of possibilities for X)
5.3 proofreading again.
5.7  "P(X > m) no larger than .05" means "P(X > m) < .05"
A)  You simulated flipping a coin with P(Heads) = .2, 20 times.  This was B(20, .2).  Using Table C, sketch the probability histogram for B(20, .2). 
You also simulated flipping a coin with P(Heads) = .6, 20 times.  This was B(20, .6).  Table C doesn't have .6.  Make a table column for B(20, .6) by exchanging the role of success and failure, and using Table C. Sketch the probability histogram for B(20, .6).
A) See below for statement of problem.

Sec.5.1 cont'd:  B and C use just the algebra rules; you can do them even if you don't "understand" the binomial, as long as you accept X=S1+S2+...+Sn., p.340 
B.)  Let S be a random variable with this distribution: 
s     1     0 
prob  p    1-p
Calculate the mean of S.(Check your answer against the calculation of  the mean of S on p. 340).  Calculate the variance of S, using the definition of variance (p. 336) .   This is the fill-in for the "hole" in the  text p. 380 --- "Similarly [?] the definition of the variance shows that sigma2S = p(1-p) " Read and understand the rest of the derivation of the mean and s.d. of the binomial, X, p. 340. 
C) Write up the derivations of the mean and s.d. of the sample proportion p-hat (p.342), from the mean and s.d. of the binomial X given on p. 340, by using the fact that p-hat = X/n.  This fills in the "apply the rules" , top of p.342. 

Read, discuss

4.81, 4.82 insurance
 Optional 
4.85 more portfolio.
 

P) Do problem A below, where variance of Y = "c" pounds (and variance of X = 1).

5.26, 27 Waiting time to first success = "geometric" distribution.  Like the "chain" email problem.


A) One more "algebra" problem:  reread #4.76 p. 309--"weight the measurements..."
Suppose I have two devices that measure "correctly" but have different variability.
Call a measurement from the first  X.  Measuring a 50 pound object, X has mean 50 pounds and Variance 1 pound.
Call a measurement from the second Y.  Measuring a 50 pound object, Y has mean 50 pounds and Variance 3 pounds.   Take one measurement from each, and take a weighted average, W = aX + (1-a)Y.  (In example 4.28, p.304, a =.2, (1-a) = .8.  In problem 4.76 (slightly different notation), a = 1/3, (1-a) = 2/3)
a) What is the mean for W?
b) What is the variance for W?  (it will contain "a")
c)  Using calculus, find the value of "a" which gives the smallest variance for W.  (Take the derivative of your variance formula with respect to "a", set it = 0, solve for a)
d) Show that your weights obey the rule described in #4.76.
(If you go through this with one more level of abstraction--Variance of Y = c times Variance of X  (c = 3 in this problem)--you will have "done" the "Statistical theory" in the book that tells what the weights should be.)

Addition:  Investment portfolio (or other "mixings" like 4.28, 4.76)  Applet  "Two-asset portfolios" aX+(1-a)Y.  What the applet doesn't do is tell you which proportion (which "a") gives a particular result, so you can't tell how to minimize your risk (s.d.).
QUIZ FRIDAY?  Computing mean and s.d. for a discrete R.V., using algebra of means and variances (pp.298, 300)

Project Presentations: 
FeiFei Chu and Wendy Lane
Alynn Kramer and Jillian Kline

Homework questions? Day23
 Diaconis article.  How does he feel?
  5.1 and 5.2 p. 351.
Binomial distribution , continued: for distribution and table use,  Day 23
If X is B(n,p) then  the Mean of X = np,  the Variance = np(1-p)   How?
    Let   X=S1+S2+...+Sn  , where each Si  is a Bernoulli trial.  Find the mean and variance of one such S, and (p. 340-1), use algebra of means and variances.
Sample proportion:  Divide number of successes X by number of trials n. 
  p-hat = X/n.    Mean of p-hat = p,  Variance = p(1-p)/n  (use algebra of means and variances, on X/n)
 
returning to Sec. 4.4  Algebra of Means and variances of R.V.'s X, Y, continued--questions?Day 23
Details  for Algebra problems: Handout and  link here
Pair up.  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
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