HW questions? Density graph.Keep your
graph
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.
HW today
A. Density-->CDF handout: Find
P(X < 0), P(X < .1), P(X < .2), ,...,P(X <
1) by calculus. (Note that it might be easier to find P(X < x0)
and plug in the different values). You have just found F(x) for x
= 0, .1, .2, .3, ..., 1. Check that it matches what you graphed last
night using square-counting.
We'll start here Friday
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.
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.
(Some) Properties: 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
For Continuous X, F is a continuous function, increasing in intervals where
f(x) is > 0.
(Mixed X, with F having some sloped and some step parts, later)
------- ------ -------
HW: 4.2 pp.111 to 114 READ
P. 118 Top.
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.
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