Math 300 , Spring 2004, Day 22, M, March 29 Hit reload...After Class

Variances of Binomial and Poisson:
Binomial X : mean np, variance npq.  Standard deviation is square root of variance. (same units as X)
Keep p constant, let n increase.  Range of possible values is 0--n.  Mean stays in same place in this scale (p of the way from 0 to n)  But the standard deviation grows not like n but like square root of n; bulk of probability takes up less and less of total possible range.  See binomial applet, let n go from 5 to 60.  Try different p's.  Bar below axis shows mean, and 1 s.d. either side.   Mean and s.d. are given numerically at the bottom of the "Distribution" column.
Poisson: Mean lambda, variance lambda. Standard deviation is square root of variance.  In the Poisson applet, lambda = ratextime. Increase lambda by increasing  rate per unit, and/or increase number of units of time.  Spread increases like square root of lambda. It's a little harder to grasp visually for the Poisson than for the binomial, because the Poisson graph is re-calibrated to keep the distribution spread looking about the same.  Look at the width of the mean+s.d.bar compared to the distance of the mean above 0. (Note the y axis doesn't cut the x axis at 0 for larger lambdas.)   Visualize for lambda = 1 (s.d. = 1) and increase to lambda = 100 (s.d. = 10)

Moral: the variability grows like the square root of the number of trials; not nearly as fast as the possible outcomes.  This allows us to find predictability in "averages".
Quiz Friday  Monday: Closed book, Expected value stuff, chosen from:

   Deriving, E(X(X-1))of Poisson. (for sure)
   Var(X)  from E(X(X-1)) and E(X)  (how to find & why)
   Proofs for Var and Cov alternate formulas.  Var(X+Y) = Var(X) + Var(Y) +2Cov(X,Y) (derivation )
   True/false: X,Y Independent <=?=>  E(XY)=0, Cov(X,Y)=0  (like Ash HW)
Ch. 4, Continuous distributions--outcome is a (continuous) measure, not a count.

Questions?  f(x) on p. 95 and g(x) on p. 96 --piecewise-defined functions

Physics: Density (of a rod...) = amount of mass per unit interval. If density varies along the length of the rod, we can talk of the density f(x) at a point x.  The mass in a short interval of length dx at that point is (density x interval-length) = f(x)dx.   Mass between a and b is sum of masses in short intervals:  very short intervals, and you have _aS^b f(x) dx.
Let mass be our metaphor for probability, area our representation for mass=probability.

Density curve:  f(x) > 0, with total area under f(x) = 1  "Probability density function" = "pdf"
Probability = Area under the curve.  P(a<X<b) = area between a and b. = _aS^b f(x) dx.    _-ooS^oo f(x) dx

We'll use integration to find areas under the curve (usually.  The Normal density can't be integrated with a formula).

--The total area under the curve being 1 forces this:  as x goes toward plus, or minus, infinity, f(x) goes to (or is) 0.
--  _aS^a f(x) dx = 0,  so P(X = a) = 0 for any continuous distribution.

4.1:  Newish from calculus: Piecewise defined functions.  You have to make sure you're integrating the right formula; break the integral at points where the formula changes.

Any function g(x) > 0 can be made into a density by dividing by a number that makes the result have total area 1.
   What number to divide by?  The area under g(x).

Example:
 f(x) =  x-2,  2< x < 4                     Graph it.
        = (x-4)²,  4< x < 5                Is it a density function?
        = 0 elsewhere.                        1 ?=?  _-ooS^oo f(x) dx
_-ooS^oo f(x) dx =  0 + ₂S⁴ (x-2) dx  + ₄S⁵(x-4)² dx + 0 =....=....= 2 + 1/3 = 7/3.
f(x) is not a density function.
But g(x) =(3/7) f(x) is!  Call the r.v. with this density function X.
  P(X< 1) = 0
  P(X< 3) =  _-ooS³ g(x) dx = ₂S³ (3/7)(x-2)dx  = ....= 3/14
  P(X > 3) = 1- P(X< 3) =  11/14
             or =  ₃S^oo g(x) dx = ₃S⁵ g(x) dx =  ₃S⁴(3/7)(x-2)dx +  ₄S⁵(3/7)(x-4)²dx

I won't try to put all  the calculus computations on the webpages--too time-consuming.  So take good notes, check with one another for "repairs" to notes.

HW:  Read 4.1.
Note:  f(x) = |x| is "already" a piecewise function:  f(x) = x if x > 0,  = -x if x < 0  See p. 96 for examples of use.
p. 103-4, #1 thru 6
Read ahead pp.104 to 114.  (You can do the following problem whether or not the reading makes sense.)
A.  Wrong graphs?  p. 96, graph fig. 2 is wrong in my text.  It shows a smooth curve, with the slope = 0 at x = 0.  Check yours, regraph and fix it if it's wrong.
Likewise p114 graph fig. 16 is wrong in my text. Graph it, and see if yours is right or wrong.  (The formula is just above the graph.)