Math 300 , Spring 2004, Day28, M, April 12 Hit reloadAfter class

Start a page of named continuous distributions, with formula, mean, variance for each.
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)

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.)
Start here Wed:
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.)

HW:
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.
   D.  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).
   E.  Find E(X) for the first Density handout, f(x) = (x - .5), 1<x<2.  Check that it is above 1.5, as the balance point must be.
F.  Find Var(X) for the Density sheet formula (f (x) = x - 1/2, 1<x<2).
Ash p. 221,
#1,
#2  Do the calculus, get the obvious answer.
#1, #3 (use p. 95)
#15
On "DENSITY PROBLEMS" handout  (from Strait textbook), Find E(X), E(X²), var(X) for #1 and #4
Ash p. 233
#2 (you did E(X) already
Additions: B. Use substitution to find the indefinite integral in E(X) for the standard normal distribution.
 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.

You must click on 'Check your answer' to make the COW aware that you have entered an answer.   Help or Hints may be useful.

The main other thing you need to know to work it is how to type the math (mostly it works as you expect):
http://cow.math.temple.edu/~cow/Manuals/TypingHelp.html

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.

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