MATH 251, Probability and Statistics I, Fall 2001, Oct. 22, Day 22

Spinning pennies updated
Independence was used last HW for :  (a) assuming independence when setting up a probability model.
This HW we will use it for (b) checking if two characteristics (factors, events) are independent (usually in the context of a two-way table)--as I did in class last time.
If we have two random variables X and Y we will say they are "independent random variables" if every event involving just X is independent of every event involving just Y.  (Usually we know this  from the setup of the probability model, like a above.)

Continuous Random variables:
We define the distribution of a continuous R.V. X using a density function f:
    Definition:  f(x) is a density function iff
        f(x) > 0, and the area between f and the x-axis is exactly 1.  Then the probability that X is between a and b is
        P(a < X < b) =  area under f(x), above the x-axis, between a and b. (the integral of f(x)dx between a and b).
Because the probabilities are areas, it's easy to see that the axioms (rules 1-4 p. 305) hold.

Discrete random variable:  P (a< X < b) doesn't equal P(a < X < b) usually;  P(X=a) can be non-zero.
Continuous random variable: P (a< X < b) = P(a < X < b).  P(X=a)=0 for any particular a.  We can only have positive probability on an interval. (discussion pp. 319-20)

For now, our continuous random variables will all be of the Normal distribution family. " Z" will usually be reserved for a standard normal variable.  Many chance mechanisms will have distributions that are approximately normal.

Mean and variance of a probability dist of R.V. X:
muX p.327   sigmaX p. 336 (X discrete) parallel to those for data. (Probability = long-run proportion)
   (Sometimes the mean of X is called the "expected value of X")
Law of large numbers:  The average (mean) of the values of X observed in many (independent) trials approaches muX .

Algebra of means and variances.
Read 4.3 and 4.4.
Hand in: 
Independence 
p. 306 
4.36 Age & cancer tests
p. 340 
4.55 rainfall
4.57 clueless gambler

Random variables--Continuous 
4.44 
4.45 c,d,e,f 
4.46 uniform 0-2
4.47 Y = sum of 2 uniform.  Compare with sum of 2 dice, the discrete analog.
4.48 normal
p. 409 (yes) 5.29 part a  coke fill (the answer in the back of my book is wrong.  Yours may or may not be.)

Means  sec. 4.4, beginning 
p. 342 4.58 "law of averages"? 
Means of discrete r.v.'s 
p. 340 4.50 Keno 
4.51 mean grade
4.52, 53 insurance
p. 343 4.73 tversky

And variances: 
4.61  s.d. of grades
4.64 mean and s.d --> r.v.
Do A and B from the Algebra of Means and Variances page.

 Read, discuss 

"Prestidigitator of Digits", on reserve + outside my door:  Moore and McCabe present the "relative frequency" foundation for probability.  The same probability rules apply to "personal probabilities," where an individual assigns fractions between 0 and 1 to the outcomes, giving some sort of "likelihood" that the outcome will happen. (they still have to sum to 1, P(not A) = 1-P(A), and P(A or B) = P(A) + P(B) if A and B have no outcomes in common.) 
--What is the capitalized name given in the article to this kind of personal probability approach? 
--How does Diaconis feel when he sits in front of his paper? 
 

 Optional 

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