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.
(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.
------- ------ -------
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. 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.
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
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.
------- ------ -------
HW Ash p. 118
#3 (the little curved piece has formula 1/3 x2)
#7
#9 (to check if it's a legal density you have
to make sure F has no jumps)
#11 Hint: move a bar across the page and watch
how the area to the left grows, in each interval.
A. On your Density 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)
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