Math 300 , Spring 2002, Day32, W, April 17 Hit reload to get most current version

f(x) = 

. There is no "closed form" formula in elementary functions for this.

The parameters are the mean and standard deviation for the normal distribution. You've been told that but we have not proved it. Last time we proved that the mean and the standard deviation for the Standard Normal distribution Z were 0 and 1.

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 different parameters.

How do we find the distribution of a function of a continuous random variable?

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).

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.

Examples: Linear transformations first. Normal, Uniform.

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.)

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