HW Try to
Finish all the HW given on Friday. Day 30
Bring questions!
Next, Ash Ch. 5 (first 5.1, 5.2)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
= = =
Functions of Random Variables, cont.: See Day
29.
New: Density method, Day 29
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.
Examples: Uniform to e, and
Lognormal, in
Excel: fX(x)dx =area in dx interval = area in dw
interval = fW(w)dw. (Dot to
dot is an interval)
Handouts from last time:
--Two sided handout, the Lognormal
distribution (W is lognormal, X = ln(W) is normal. W = eX)
Skewed data often "turns normal" when you do a
logarithm to it. If so, the original data is
"lognormal."
This handout has homework problems A-D in the corner.
Other details:
-- One sided handout, Changing variables when there's a Y = -X type
situation. That is CDF method.
Density, just drop the minus in the derivative (Day 29, additional technique c). Try
it with the handout W = -X normal.
When the transformation function is not 1-1, two (or more) x's
map onto the same y. (Day 29,
additional technique d)
The Lognormal
handout, backside, bottom, models the Density method.
The CDF method may give a double inequality, to evaluate
using the X distribution..
Cf.
6-sided die, W = (X-3)2. FW(w) = P(3 -
sqrt(w) < x < 3+ sqrt(w)) Day29
Applications (lognormal or other transformations): Xeroxed
pages outside my door, "Re-expressing...."
Just reading the NY Times this
weekend I noticed two highly
skewed distributions; half of the $$ given to a charity tend to
come
from 10% of the donors. Half of the revenues to gambling casinos
come
from 5% of the gamblers. The numbers are wrong but the
ballpark idea
is there. The "80-20" rule,
which I learned this way: The first 80% of a job can be done in
20% of
the time; the remaining 20% will take the other 80% of the time to
acheive
completeness or perfection. How
(not) to use it. If something is worth doing, it's worth
doing badly...
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