Math 300 , Spring 2008, Day 25, M, Mar 31 after
class
Hit
reload ..
Quiz today, first. Bring it to
my office, get exam back + solutions; then we'll start class when
all are finished.
Closed book, Expected value stuff, chosen
from:
(for
sure) Deriving,
E(X(X-1))of Poisson.
(one
from) 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 )
(one from) True/false: X,Y Independent
<=?=> E(XY)=0, Cov(X,Y)=0 (like Ash HW p.223 # 14)
HW: Read 4.2
pp.108 to 114, READ P. 118 Top.
Density-->CDF handout
Find the formula of F using calculus, and check that it matches
your graph. Solutions
151 Densities to CDF Finish finding all
the F's formulas using calculus, and check that they match your graphs.
Ash p. 118
#4, 5, 6
(general facts)
#2 (uniform)--the formula is on p. 106
#10a,b by calculus--sketch both f and
F
#10c: Draw f. Find F(-.1), F(.1), F(.5)
by geometry. Connect those points to get the graph of F
(Use the idea behind the uniform derivation pp. 106-7).
HW read rest of 4.2
Postpone the rest
Ash p. 118
#11 f to F: Hint: move a bar across the page and
watch
how the area to the left grows, in each interval.
#3 P's from F (the little curved piece has
formula 1/3 x2)
#7 (mixed)
#9 (F to f: to check if it's a legal density you
have
to make sure F has no jumps)
A. On Density-->CDF
handout ,
for x =1, 1.1, 1.2, 1.3, ....1.9,
verify that [F(x + 0.1) - F(x)]/0.1
= f(x+.05) (f at the middle of the 0.1-wide interval)
= = = = = = = = = = = = = = = = = = = = = = = = = = =
4.2: Cumulative Distribution Function
(CDF,
"Distribution Function") cont'd
F(x) = P(X < x).
Defined for every x on the real line. Capital
letter.
Continuous:
Area to the left of x under the density.
F is a Continuous function. Think
of moving x from left to right across the density curve, tabling
the area to the left of x. Since the area increases bit by bit as
x increases, F will be a continuous function, even if the density f is
not.
HW questions?
Day 23
Counting squares: Two Handouts.
Density-->CDF handout
, Solutions
Computation: F(x0) is
the area under the density curve from minus-infinity to x0.
Find it by taking the definite integral.
If f is defined
piecewise,
F will be also, and the computation of area may need to piece
together
separate integrals.
After you have found the
formula(s),
you can drop the subscript from the x.
Nit-pick with Ash: p. 112
ff--she uses x both as the variable of integration f(x)dx, and the
limit
value (integral to x).
Mathematicians regard
that as bad grammar. If you use x inside in f(x)dx, use x0
as the limit value.
If you want to use x as the limit value, use f(t)dt inside.
Examples: from 151
Densities to CDF
A, "Uniform" Since f is constant
through
the interval [0, 1], F will increase at a constant rate through
the same interval.
So
F(x) is a straight line, rising from 0 at the beginning of the interval
to 1 at the end of the interval.
F(x) = 0, x< 0
= x, 0<x<1
= 1, 1< x
For uniform distributions
over other intervals, say [a,b], F is still the straight line going
from
(a,0) to (b,1)
Formulas p. 102-3,
106-7 (You can get F as area under f between a
and x by geometry (Ash) or by calculus;
or by high school techniques for finding the line between 2 given
points.
Know one way.)
B, "triangular"
f(x) = x,
0<x<1
0<x<1, F(x) = 0Sx t dt = [ t2/2]0x
= x2/2
2-x, 1<x<2 1<x<2, F(x) = 0S1
t dt +1Sx(2-t) dt = 1/2 + [2t- t2/2]1x
= 1/2 + [2x- x2/2] -[ 2- 1/2] = [2x- x2/2]-1
Checking endpoints of intervals where each formula holds shows that it
is continuous.
(Some) Properties: P(a
< X <
b) = F(b) - F(a)
F(x) --> 1 as x --> infinity (it may actually get to 1, earlier)
F(x) --> 0 as x --> minus infinity (as x decreases) (it
often gets
to 0)
F(x) is NONDECREASING (it may be flat or increasing as x increases, but
it's never decreasing)
F(x) is flat in intervals which have no probability
For Discrete X, F is a step function--jumps at the lumps. P(X = a) =
size
of jump at a.
For Continuous X, F is a continuous function, increasing in intervals
where
f(x) is > 0.
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Start
here Wed: Mixed distributions--some
parts continuous, some parts discrete (jumps).
Example: X = Lightbulb life for a bulb
that burns
10 hrs/ day, then is turned off, turned on again the next morning.
Bulbs often burn out at the "moment" of being turned on. But have
a chance of burning out while on.
Density
inadequate,
discrete distribution inadequate.
"Pseudodensity" puts "lumps" in places ("delta function"--finite
nonzero
area attached to a single point.)
Examples: Uniform from
0 to
10, P(10) = .5. (Flip coin: win $10 if H; if T, spin
spinner labeled 0to 10 for winning.)
Old lightbulb (900 hrs already)--Turn on,
then
burn till it burns out.
Cumulative distribution function is completely
adequate--increases in continuous regions and jumps at lumps.
- - - - - - - - - - - 
Graph reading for probability:
The vertical axis "measures" probability; the distance between two
points on it represents the probability between the two corresponding
points
on the horizontal X axis.
- - - - - - - - - - - - - - - -
From F(x) to density
(continuous):
F'(x) = f(x). (If
F(x) has no derivative, a "corner", f will change formula there)
(Caution:
F(x) is NOT the antiderivative (indefinite integral) of f(x).
It's the definite integral from minus infinity to x.)
Fundamental Theorem of the Calculus: one
version
(footnote,
p. 115)
Let F(x) be
the area under the curve f(x) between a and x. Now consider how
F(x),
the area, changes as x changes.
F'(x) = The Rate
of change of F(x) as x changes:
Let x go from x0 to x1
. The area changes from F(x0) to F(x1).
The area is approximately the rectangle f(x0)(x1 - x0 ) .
The rate of change is the change in F,
F(x1) - F(x0),
divided by the change in x, x1 - x0 .
This is approximately f(x0)
(if f is continuous at x0).
As x1 - x0, the
"approximately"
disappears.
The slope of the CDF F is the height
of the pdf f.
From F(x) to density (discrete):
P(x) = size of jump in F at x.
From F(x) to pseudodensity (mixed):
combine F'(x) where it exists, with discrete jumps.
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