Math 300 , Spring 2004, Day 31, M, April 19 Hit reload...After Class

1/6 1/6 1/6 1/6 1/6 1/6
 V   V   V   V   V   V     x    (these are supposed to represent the lumps of probability)
 1   2   3   4   5   6

Y = X-3
1/6 1/6 1/6 1/6 1/6 1/6
 V   V   V   V   V   V     y     Every x point maps to a y point, and drags its probability along with it.
-2  -1   0   1   2   3

W = Y²                                     Every y point maps to a w point,
1/6 2/6         2/6                 1/6       and drags its probability along with it.
     V           V
 V   V           V                   V   w     Sometimes two points map to the same point.
 0   1   2   3   4   5   6   7   8   9

First step always: Determine which regions of y (or w or whatever) have no probabality by looking at range of x's, what happens in transformation.  Above, 1< x <6, so (1-3)< y <(6-3).   w>0 because it's a square.

Density method:  In the continuous case, Areas get dragged along from one axis to the other. Mark off a bunch of values on the x-axis.  (Assume first that Y = g(X) is 1-1)

x_i -->y_i , x_i+1 -->y_i+1 ,  Width of the interval in x is dx =  x_i+1 - x_i ,
                         width of the interval it maps to in y is dy =  y_i+1 - y_i
But the Area above that interval in x has to be the same as the area above the corresponding interval in y:  Each area is approximately a rectangle:  (This is analogous to the lumps of probability in the discrete case)
  f_Y(y) dy = Area = f_X(x) dx.  Solve for  f_Y(y), and use y = g(x) or x = g^-1(y) to get everything in terms of x.

CDF method:  Look at the discrete case, W = (X-3)²  Start with wanting F_W(w_o).  = P(W < w_o)
What X-interval maps into W < w_o ?   (X-3)² < w_o so  -sqrt( w_o) +3 <  X < sqrt(w_o)+3.
Find that probability using the distribution of X.  P(-sqrt( w_o) +3 <  X < sqrt(w_o)+3. ) =
    sum of 1/6'ths for all values in this interval.  Or =
      P(X < sqrt(w_o)+3) -  P(X <  -sqrt(w_o)+3)
          For this discrete case, list the values of w and calculate these probabilities.

W = (X-3)²
1/6 2/6         2/6                 1/6
     V           V
 V   V           V                   V   w
 0   1   2   3   4=w_o5   6   7   8   9

1/6 1/6 1/6 1/6 1/6 1/6
 V   V   V   V   V   V     x    If w_o = 4, (-2+3)< X < (2+3), by inspection.
 1   2   3   4   5   6

Complexities in the continuous case:  You may sometimes be able to work directly with a formula for F_X.  Great! do it. If there is no explicit formula for F_X (e.g. the normal distribution and friends) you'll need to mess with integrals, either changing variables or using the Fundamental theorem to get the f_Y  or f_W  you want.

Figuring out what x's map to a y, or what ranges of x's map to a range of y.  or what dx's map to a dy.
A graph of y =g(x) may help, going over and down from a y to see what x('s) map to it. (Ash's favorite method)  Think of the probability density for the x's piled along the x axis, the probability for the y's turned sideways and piled along the y-axis.

So that's what I should have done....
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Mini-exam due today.
No class Wednesday.  You're welcome to come in and work on HW together...

Functions of Random Variables:  See Day 29.
Old Handouts:
--Two sided, the Lognormal distribution (W is lognormal, X = ln(W) is normal.  W = e^X)
    Skewed data often "turns normal" when you do a logarithm to it.  If so, the original data is "lognormal."
          More about application next time.
    This handout  has homework problems in the corner.
-- One sided, deals with someY = -X questions.
New handout: changevariables.xls
--You can think of the transformation as shifting points, and chunks of probability, from one horizontal axis to another.  Or graph the transformation relation and follow the shift from a horizontal axis to a vertical axis.  Try to get used to both ways.-

HW:  I think this is the easiest sequence.  You may feel differently.  Feel free to vary.
Do whatever you haven't done.  Try doing Density method first.
Do B from the handout (Discrete) (to hand in Friday)
 Then do (B, C,, parts a,b,c)  D from Day 29.

Then (optional for the weekend) do D from the handout (any method the CDF method) (to hand in)
- - - - - - - - - - -
   (Day 29 B,C, parts d), D from the handout by the Density method if you didn't do it by the CDF method.
Then do A and C  from the handout (to hand in)
#1, 3, 5 from the text are more complicated because the functions are defined much more piecewise.  Work on them, and use the answers, till you understand them..