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.
Counting squares: Two Handouts.
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 handout:
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.
------- ------ -------
HW: 4.2
pp.111 to 114 READ P. 118 Top.
Density-->CDF handout: Graph CDF by counting squares, carefully.
"151" handout: Finish finding all the F's using calculus, graph
using tables and check with formulas for F.
Ash p. 118
#4, 5, 6
#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).
Handout "Density problems"
Find F(x) for #2. Then take its derivative
and see that you get f(x) back again.
Monday: 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. Another: Old lightbulb (900hrs already)--Turn on, then
burn till it burns out.
Cumulative distribution function is completely
adequate--increases in continuous regions and jumps at lumps.
Getting density from CDF (continuous)
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