MATH 251, Probability and Statistics I, Fall 2005, Mon. Oct. 24, Day 25After class

Read Sec. 5.1.  I've covered 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).
QUIZ WEDNESDAY or FRIDAY:  Computing mean and s.d. for a discrete R.V., using algebra of means and variances (pp.298, 300)
Hand in: 
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. 

Postpone the rest: 
p. 351ff 
5.9b random numbers
5.5 proofreading, mean and s.d. Also, graph p(1-p) for 0< p <1 to reinforce your understanding of part c. 
5.12 random walk stocks (For part b, calculate the values using the formula!)
5.22 trip marketing
5.24 binomial coefficient facts.  (Not sure they're true?  Check with your computations in #5.12 for plausibility. Remember in any binomial situation you can reverse the roles of Success and Failure.  Note b and c together give nC1 = n.

Normal approximation
5.17 alcohol study (this reasoning is that of significance tests, coming up)
5.23 proportion of sample
5.25 variability on exam

Read, discuss  Optional 
P) Do problem A below, where variance of Y = "c" pounds.
Postpone the rest:
5.26, 27 Waiting time to first success = "geometric" distribution.  Like the "chain" email problem.

Q) Calculate the "finite population correction" for n = 25, and N (population) = 250 (10n, some texts' rule of thumb). And for N = 500 (20n, IPS's rule of thumb). 

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.)

Homework questions?  Diaconis article.  How does he feel?
  5.1 and 5.2 p. 351.
Sec. 4.4  Algebra of Means and variances of R.V.'s X, Y, questions?Day 23
Pair up.  If you weren't here Friday, be sure to be with someone who was here.  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.   Mean is 3
::Find the mean and standard deviation of  their average, Xbar = (X1 + X2 + X3)/3
Details and solutions for Algebra problems link here
Start here Wed:  Addition:  Investment portfolio (or other "mixings" like 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.).

Sec. 5.1  X Binomial   B(n, p) , continued  Moved to Day 26
 

Next--5.2: Sampling distribution of the Mean of an SRS


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