Application of discrete dist: Bird calls. (Due today) What does it say, that the geometric distribution describes the bird's song length?
Quiz Monday: Closed book, Expected value stuff, chosen from:
Deriving E of Poisson, Var of Poisson from E(X(X-1)) and E(X),
Proofs for Var and Cov alternate formulas. Var(X+Y) = (derivation)
Independence and E(XY), Cov(X,Y)
Ch. 4, Continuous distributions--outcome
is a measure, not a count.
Questions? f(x) on p. 95 and g(x) on p. 96 --piecewise-defined
functions
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.
We'll use integration to find areas under
the curve (usually. Normal density can't be integrated with a formula).
The area under the curve being 1 means that as x goes toward plus, or minus, infinity, f(x) goes to (or is) 0.
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).
I won't try to put the calculus computations on the webpages--too time-consuming for me. So take good notes, check with one another for "repairs" to notes.
4.2: Cumulative Distribution Function
(CDF, "Distribution Function")
F(x) = P(X < x).
Defined for every x on the real line.
Discrete: Sum of probabilities to the
left of x (including x). Jumps at each lump. "Step function."
Continuous: Area to the left of x under
the density. Continuous function.
Capital letter.
------- ------ -------
HW: Add Hypergeometric variance to your distribution sheet.
Read 4.1.
p. 103-4, #1 thru 6
(Graphing: If you prefer, you can use Excel,
or even SPSS, to calculate and graph a collection of representative points.
Then you can draw the connecting lines on the graph by hand. Even
if you remember how to get the computer to get a smooth connecting line,
neither package can deal with piecewise formulas.)
Read ahead pp.104 to 114. (You can do the
following problems whether or not the reading makes sense.)
A. Normal Distribution: Use the
table for the Standard Normal Distribution and plot P(X <
x) vs. x, for x = -3, -2.5, -2, -1.5, etc. by .5's, to +3. Connect the
dots with a smooth line, and you will have graphed the CDF of the Standard
Normal. (There is a table on p. 129, Ash--"Unit normal")
B. p. 114 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)
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