Math 300 , Spring 2002, Day 27, F, April 5 Hit reload to get most current versionAfter class

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.
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:  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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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. 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.   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.

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.
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Will be assigned Monday:
#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.