Mini-exam due Wed or before.
After class: guide to what's doable with
CDF method:
Reading: try reading the handouts, any parts with F's (or G's) in them
(CDF method). This is everything but the backside of the lognormal
handout.
DO C below, and A and B, parts a,b,c (CDF
method).
From Lognormal handout, Try A. Do B by
the CDF method. D you can try by CDF if you like; method is optional
there.
HW from day 29: 4.6: Read as much of
4.6 as you can stand + some more. I have
misprints: p.142, bottom formulas are fY and FY,
not X as I have. The axes in the graphs on p. 148 are
rubbery.
A) X is uniform on [0, 1]. Let Y = cX + d. (assume c
> 0) a) On what interval does Y have positive
probability?
b) Write FY(y0) = P(Y < y0),
substitute Y = cX + d, and solve to get P(X.....) Find a formula
for this, using the CDF for X. (no integrals needed. Use
technique (a) below. ) Answers? Check by drawing a picture,
or pictures, compare with Ash pp. 106-7.
c) Take the derivative with respect to y, thus finding fY(yo).
For F and f, be sure to give the intervals where your functions are in
force.
d) Now use the Density method to go straight from fX(x)
to fY(y). Sketch it for c = 2, d = 3, and show where
x= 0, .5, and 1 map to.
B) Repeat A for the uniform distribution, only
assume c < 0 (so values flip direction.)
For c = -2, d = 3, show where x= 0,
.5, and 1 map to, before you begin. Use the CDF method, then try
the Density method (use technique (c) for coping with the minus.)
. Instead of continuing with the change of
variables, Take the derivative of this with respect to z (drop the
0-subscript to do so). HW: Problems from
lognormal handout. Hand in
A, B, C, D as you finish them. B may be easiest to start with.
Ash p.156 #1, 3, 5 are more complicated because the functions are
defined much more piecewise. Work on them, and use the answers,
till you understand them.. HAND IN
3, 5
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Functions of Random Variables: See Day 29.
Handouts:
--Two sided, 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."
More about
application next time.
This handout has homework problems in the
corner.
-- One sided, Changing variables when there's a Y = -X type
situation.
Today (Friday) only got thru CDF part of Day 29 page. Density method next time.
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: area in dx interval = area in dw interval. (Dot to
dot is an interval)
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