Ch. 4, continuing Sec.4.4, 4.6. (4.5
optional--does a "confidence interval"!),
How do we find the distribution of a function of a continuous
random variable? continued.
Last time: W = aX +b, for X N(mu, sigma), and a >0
Y = g(X). Two methods:
1) CDF method: Find FY(y0) = P(Y <
y0) = P(g(X) < y0)
Do what you need to, to rewrite this as P(X <?
g-1?(y0) ) (If
g doesn't have an inverse over the whole range this could be more complicated.
And the inequality might reverse.)
This can (usually) be written as an integral with upper limit x= g-1 (y0), f(x)dx. Change variables to y, inside the integral and in the limits, until you get an integral with y0 as the upper limit. That's your FY(y0).
2) Density method: Suppose X has density fX(x). We
want the density of Y, fY(y). A thin rectangle at xo
of width dx and height fX(xo)gets mapped by y = g(x)into
a rectangle at yo of width dy and height fY(yo).
(If g is one-to-one, that's all that gets mapped into the new rectangle.)
Set the rectangles equal: fY(yo)dy = fX(xo)
dx
Solve: fY(yo) = fX(xo)
dx/dy. You have to find dx/dy, and change variables to get everything
in terms of y.
Since y = g(x), dy/dx = g'(x). You can put it under 1 to get dx/dy.
Or find x = g-1 (y) and take the derivative of x with respect
to y.
Additional techniques:
If the transformation function is not 1-1, you'll
have two (more?) x-chunks mapping onto the same y. Be sure to account
for both.
Density method--If the transformation function
"flips" things (x1< x2 , but y1 > y2),
dy/dx will be negative. Just throw away the negative (take the absolute
value.)
CDF method--Instead of changing variables inside the integral, once you get a formula, you can find the density by taking the derivative. This requires a more sophisticated use of the Fundamental Theorem of the Calculus, with the chain rule.
Practice here: http://www.math.temple.edu/~cow/If the CDF has a nice formula, you just substitute into IT, not the bound of the integral .
Calculus Book II > 1.Integration > 4.Fundamental Theorem > 1.Differentiation and the Fundamental Theorem
Read the Help, then try problems. Try these: 1, 2, 8, then any you like. Type x3 as x^3, sqrt(x) or x^.5.
If X does not have the same formula everywhere,
you'll have to follow each piecewise formula in the transformation.
HW: Read as much of 4.6 as you can stand.
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 this using
the CDF for X.
c) Take the derivative and find fY(y0).
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.
(With Day 34) Ash p. 156 #1, whatever
method you like. Also Sketch the density of X and the density
of Y, and show where x = 0, 2, and 5 map to.
B) Practice the Fundamental Theorem using the Temple U. COW system
above.
| Sievers home |
Math300-Sp02/Day33.htm
|
8:30pm
|
4/21/02 |
This page belongs to Sally Sievers who is solely responsible for its content. Please see our statement of responsibility.