MATH 251, Probability and Statistics I, Fall 2005, Mon. Oct. 17, Day 22corrected

Read 4.3, all. Begin 4.4:first mean pp. 291-3 and variance/s.d. p300-1; then the rest.   
Hand in:  Sec. 4.3
Random variables--Discrete 
4.40 discrete or continuous? (c: debatable?)
4.41 nonword errors Also make a probability histogram of the distribution.

Discrete: building more complex models
4.48 sum of 2 dice  Hints:  Think of a red die and a black die. 
A two-dimensional graph is a good way to arrange this: 
on the horizontal-axis mark the possibilites for the red die (1,2,3,4,5,6) , 
and on the vertical axis mark the possibilities for the black.(1,2,3,4,5,6). 
Then each integer pair, intersection on the grid,  stands for a possible pair;  (1, 3) means 1 on the red and 3 on the black.  Note the geometry of which pairs have  equal sums.
4.49 regular +0-6 die Also graph it.

4.52 3 student reps-->#opposed. For part a, I suggest making a tree of the possibilities/probabilities.  The first branching will be for student A: support or opposed?  Once that is decided, we branch for student B: support or opposed?; likewise for student C.  What assumption do you make to find the probabilities?       (Also, read 4.51c, which applies here also.)
4.47 Chained letter stopping time

Continuous
4.42 pregnancy (Normal)
4.56 triangle distribution
4.55 Uniform on 0-2

4.57 mean # of friends
4.58 proportion of voters

Sec. 4.4--Mean and variance
4.60 typing errors again Find means and also variances.  Write your results down, in the book; you'll want them for 4.64 later. 		Read, discuss 
 4.53<,<
 Optional 
Random variables--Discrete 
p. 287 4.44, 4.46 (MOTS, more of the same)


Sec. 4.4
4.59 (MOTS)

Homework questions?
Sampling distributions: your data. (overheads)
    Notice x-bars and p-hats roughly normal looking, centered around population parameter.  Distribution of x-bars is narrower than distribution of individual x's. 
Actual sampling distribution for the 4- grades:  Solutions manual.

Sec. 4.3:  Often the sample space is naturally expressed in numbers, thus
Random Variable:  (X, Y, W, Z...) Variable whose value is a numerical outcome of a random phenomenon.
 Probability distribution of X tells us what values X can take and how to assign probabilities to them.
    If X has a finite number of possible values (Discrete distributions), nothing new except notation.
     P(X < 2) is "Prob. that X is less than 2."
Flip coin twice. R.V. X = number of heads:  Distribution given by table.
             x| 2  |    1    |  0 |
       P(X=x) | .25|   .50   | .25|  P(X > 1) = ?   Words:  Prob that # heads is > 1
                                     P(X = 2) = ?         Prob that # heads is 2
  Visually, "probability histogram": area = 1
<>A "Random variable"  is more complicated than a calculus variable because the variable carries the "baggage" of probabilities attached to values it can take on. (Some people dislike the name "variable" but we are stuck with it because of history.  It parallels the name "variable"  in data for some aspect of a population that we measure.)

Random variables with intervals of outcomes ("continuous")
If the sample space is an interval of values (or the whole line), the way we assign probabilities to events is with a density curve (cf. Day5,<> densities)
 (Remember density curves were idealizations of histograms--of repeating the "experiment" many many times.    
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. 262) hold.
            Choose 1 from a population:  P(a<X<b) = proportion of population between a and b 
  P(a <  X < b) = the probability that X is between a and b  is the area under the density curve, between a and b.
We declare P (X = a) = 0 , so P(a <  X < b) = P(a < X < b)
(Discrete:  P(a <  X < b) doesn't = P(a < X < b) usually.

Notation: Use capital letter for the random variable, the "label" of the phenomenon.  Use small letters for particular values it can have.  But this rule is often broken--Moore uses x-bar where many would use X-bar.

We can use graphs with the sample space on the horizontal axis, and a curve or  histogram above, to model the probability distribution.  The area above an interval then represents the probability of the interval. (The whole area = 1)  
Probability distribution:  Defined using: Graphic representation:  (Area under it is 1)
    Discrete distributions  table (sometimes formula (later)) "probability histogram"  (Other texts: vertical bars)
   Continuous distributions (formula)   "density curve" 

Sec. 4.4  Mean and variance of a probability distribution of R.V. X:
mean µx  p.292 , variance sigmax2 & std. dev. sigmax p. 300 (X discrete) parallel to those for data.  
(Sometimes the mean of X is called the "expected value of X")
Weight each value by its probability (Probability = long-run proportion) .
Discrete  µx = sum (xi pi)    sigmax2 = sum[ (xi - µx )2 pi]    (Calculating for continuous: Math 300)
        x| 2  |    1    |  0 |
  P(X=x) | .25|   .50   | .25|
µx = 2(.25) + 1(.5) + 0(.25) = .5 +.5 = 1   Mean is still the balance point.
sigmax2 = (2-1)2 (.25) + (1-1)2(.5) + (0-1)2(.25) = .25 +.25 = .5.  sigmax = .71
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