Math 300 , Spring 2004, Day 29, W, April 14 Hit reload to get most current version

Got to here Wednesday.
Return to Ch. 4, Sec.4.4, (4.5 optional) 4.6.
Normal distribution

 f(x) = . There is no "closed form" formula in elementary functions for F(x).

The parameters are the mean and standard deviation for the normal distribution. You've been told that but we have not proved it. We just proved that the mean and the standard deviation for the Standard Normal distribution Z were 0 and 1.  The proofs from the general formula "become" the same proofs if you make the change of variable (substitution) and change the limits of integration too. (Pp. 224-26)
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If we have a Normal variable X, and we "standardize", you've been told to take on faith that you get a "standard normal" variable Z--the same distribution but with 0, 1 as parameters.  Now we'll prove it.

How do we find the distribution of a function of a continuous random variable? (Sec. 4.6)

Y = g(X). Two methods:

1) CDF method: Find F_Y(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 F_Y(y₀).
Example: Normal distribution, standardize it. .= g(X).
 F_Z(z₀) = P(Z < z₀) = P(g(X) < z₀) = P( ) = P() =

   Change variables to ,  and you'll get  We recognize this is the CDF of the standard normal.  You can take the derivative to get the pdf.

2) Density method: Suppose X has density f_X(x). We want the density of Y, f_Y(y). A thin rectangle at x_o of width dx and height f_X(x_o) gets mapped by y = g(x)into a rectangle at y_o of width dy and height f_Y(y_o). (If g is one-to-one, that's all that gets mapped into the new rectangle.)

Set the rectangles equal: f_Y(y_o)dy = f_X(x_o) dx

Solve for f_Y(y_o):
              f_Y(y_o) = f_X(x_o) 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.

Example: Linear transformations first. Normal.
Let X be .  Let .  A rectangle at x_o of width dx and height f_X(x_o) gets mapped by into a rectangle at z_o of width dz and height f_Z(z_o). f_Z(z_o)dz = f_X(x_o) dx      f_Z(z_o) = f_X(x_o) dx/dz.

, so .

f_X(x_o) = , rewrite in terms of z, not hard.
Then f_X(x_o) dx/dz =, the pdf for Z !

Additional techniques:
(a)  CDF Method:  If the CDF has a nice formula F_X(x), you can just substitute into F_X(x) from P(X < something ), getting F_X(something). If F_X does not have the same formula everywhere, you'll have to follow each piecewise formula in the transformation.

(b) CDF method:  If you end up with an integral, instead of changing variables inside the integral,  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, since the upper limit of the integral will not be simple.

Practice here: http://www.math.temple.edu/~cow/
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.

(d) If the transformation function is not 1-1, you'll have two (or more?) x-chunks mapping onto the same y.  Be sure to account for both.

HW: Normal review, Ash p. 136 Note, Ash uses X*, instead of the more conventional Z, for std. normal.
Do the bold ones, read the others to make sure you can do them.
# 1, 2, 3, 4, 5, 6, 7, 9, 17, 14(X = # who show: It's binomial--use Normal approx. to binomial.)

A) Practice the Fundamental Theorem using the Temple U. COW system above, doing 1, 2, 8. "Help" will explain the topic well.  You should check your "fundamental theorem" answers to these 3 by doing the integration shown, then taking the derivative.  (For instance, in 8, integrating 2t gives the indefinite integral t².  Evaluating between x² and 2 gives (x² )² - (2)²= x⁴  - 4.  Taking the derivative gives 4x³.  Your "fundamental theorem" answer must match this one.
Then do #10.  Then read the Help to know how to attack problems where both limits of integration move, and do #9.  Practice a few more, from anywhere in the list of 20.
Do the above problems for Day29 homework.
  I suggest printing out this page for reference for the next lecture(s)
4.6: Read as much of 4.6 as you can stand. I have misprints: p.142, bottom formulas are f_Y and F_Y,  not _X as I have.  The axes in the graphs on p. 148 are rubbery.
B)  X is uniform on [0, 1].  Let Y = cX + d. (assume c > 0)   a) On what interval does Y have positive probability?
b)  Write F_Y(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.  Use technique (a) above.
c) Take the derivative with respect to y, thus finding f_Y(y_o).  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 f_X(x)  to f_Y(y).  Sketch it for c = 2, d = 3, and show where x= 0, .5, and 1 map to.

C)  Repeat B,  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 technique (c) above.

D) Applying technique (c) above: After the fundamental theorem --chain rule version--is understood, look at the derivation above for standardizing X normal.  We got F_Z(z_o) = .  Take the derivative of this with respect to z (drop the 0-subscript to do so).
You should get, after everything is done, the formula for f_Z(z).

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