Math 300 , Spring 2008, Day28, M, April 7Hit reload.after class.

Start a page of named Continuous distributions, with formulas, mean, variance for each.

HW finish these from Day 26-7 (read 4.3, including Gamma)  Ash p. 125-7 Exponential (thru #5) +Gamma &Exp
#1.
#2 (for e, you can use your Poisson table.)  #3e (cf. the bird calls sheet),  #4
#5
#6 "by time t" = "in t more minutes"
#7 [part f should say 4 particles--my book says 3.  My book also has 210 instead of 10 in the answer to g]
#8  Gamma
A and B should have been done.
C.  Substitute n = 1 in the Gamma formula for f, and check that you get the exponential (the exponential is the special case of the Gamma).

HW: Reorganized to put harder problems later
Expected value and variance, back to Ch. 7 pp. 213-223, 225-232.  Useful integrals p. 95
   From day 27 Expected value:  Read pp. 213-16.
   A.  Ash finds the mean (expected value) for the Uniform distribution on [a,b]  on pp 214-15. Understand that and check that the mean is the balance point on the graph (p. 103).
   B.  Find E(X) for the first Density to CDF handout, f(x) = (x - .5), 1<x<2.  Check that it is above 1.5, as the balance point must be.
C.  (new) Find Var(X) for the first Density to CDF formula (f (x) = x - .5, 1<x<2).
On "DENSITY PROBLEMS" handout (from Strait textbook), Find E(X), E(X²), var(X) for #1 and #4

Ash 7-1 p. 221 annotated,reordered. 1, 3 are hard integrations without p. 95.
#2  Sketch the graph, Do the calculus, get the obvious answer.
#1, #3 (use p. 95)
#15

Ash 7-2, p. 233
#2 (you did E(X) already, above. Only hard thing is simplifying the polynomials. )

Yes: HW on integration: A.  Review and practice integration by substitution at
http://cow.math.temple.edu/~cow
Choose Calculus Book II > 1.Integration > 2.Indefinite Integrals >2.Substitution - change of variables,
for indefinite integrals.

B. Use substitution to find the indefinite integral needed in E(X) for the standard normal distribution--mean 0, s.d. 1. (Ash calls it--nonstandardly--the "unit normal" ) (formula is in Fig. 2, p. 128).

C. Changing variables (substitution) in a definite integral is something we'll need. You can practice this at
Calculus Book II > 1.Integration > 3.Definite Integrals > 2.Substitution methods
About this module, and Help, give math help with the topic.

= = = = = = = = = = = = = = = = = = = = = = = = =
Questions on HW?

4-3: finish Exponential distribution Waiting time till first arrival in a Poisson phenomenon. Expect lambda per unit interval.
F(x) = 1- exp(-lambda x), f(x) = lambda exp(-lambda x), both for (x>0)
Gamma distribution: Waiting time till n'th arrival.

Expected values and variance:  (back to ch. 7)  Day 27
Instead of sums, we have integral signs.
E(X) is still the balance point on the f(x) graph.
"Law of the unconscious statistician"--
If Y = g(X), then E(Y) = integral over all y's of y times density of Y, OR
                              = integral over all x's of g(x) times density of X.
   So E(X²) is the integral of x² times f(x) dx.
Var(X) =  E(X²) -µ² still.  (proof)

Algebra of expectation rules still hold:  E(cX)=cE(X),
E(Y + W) = E(Y) + E(W)  (If Y and W are both functions of some common X, you can see this simply because integration distributes over sums.  If not, we'll have to wait for chapter 5)

Will do these Wed.
Find E(X) for exponential (p. 214).  Requires integration by parts (or Ash's handy formula p. 95).
Find Var(X)= E(X²) - µ² for exponential (p.226, and p. 95)
Find E(X) for Normal (0, 1). (not hard.)
Find Var(X) for normal (0,1), integrating by parts. (We'll use the fact that area under f(x) = 1, which we haven't proved.)

next we'll return to Ch. 4, Sec.4.4, 4.5 optional, 4.6.