Math 300 , Spring 2004, Day 25, M, April 5 Hit reload to get most current version

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.
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 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.
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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.
 

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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 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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HW   read rest of 4.2
Ash p. 118
#11 Hint: move a bar across the page and watch how the area to the left grows, in each interval.
#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)
A.  On your first 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)