In statistics, what is the "induced probability" in contrast to? All I
can find are definitions for induced probability - I understand the
definition, but I don't understand the adjective "induced". Is there
any such thing as a probability that isn't induced? What would it be?

Hi, danae05-ga:

The adjective "induced" usually refers to a real-valued function
defined on a probability space, a situation that we also call a
(real-valued) random variable.  The real-valued function then induces
a probability distribution on the real numbers, while the "original"
probability space may be something quite different, or even if it too
happens to be a probability distribution on the real numbers, that
distribution will be different from the induced probability
distribution except in special cases (e.g. the function is the
identity).

The "induced" probabilities are those associated with the outcomes of
the random variable, or simpler terms the values of the real-valued
function.  This serves to distinguish them from the probabilities
associated with the underlying probability space.

An example may be helpful by way of illustration.  Probability spaces
come in a variety of forms: discrete, continuous, and sometimes a
mixture of both!  Usually we are motivated to drag in the "induced
probability" terminology when the underlying probability space itself
has some topological structure, so that we might speak of the
random-variable as being a continuous function (or not) from the
probability space to R, the set of real numbers with the usual metric.

Let's consider a dartboard which is a circle of unit area as our
probability space.  On the dartboard are painted several concentric
circles, each having a different number of "points".  The points
scored by a dart is our random variable, i.e. a real-valued function
whose domain is the probability space (dartboard).

Now in the underlying probability space, which is continuous, the
"events" are measurable subsets of the dartboard, and the
probabilities associated with these events are their areas (since I
normalized the entire dartboard to have unit area).

On the other hand the real-valued function takes on only discrete
values, so this is not a continuous function.  Nonetheless we may
speak of the "induced probabilities" on R of a discrete probability
measure, which associates to each point value defined by the dartboard
the area of the ring (or bullseye) associated with it (recalling that
each region has a different point value).

A more sophisticated example might take a real-valued differentiable
function F(x,y) defined on the unit square [0,1]x[0,1].  Make the
underlying probability space be that unit square, say with uniform
probability density function for simplicity.  The induced probability
density function can then be related to the derivatives of F.  But in
any case the induced probabilities are those connected with the
chances of real numbers that are "outcomes", induced from applying F
to "events" (measurable subsets) in the underlying unit square
probability space.

For a more complete recapitulation of the "induced probability"
basics, see this fast paced summary, esp. the section on induced
probability density functions toward the end:

[Continuous random variables]
http://tableau.stanford.edu/~lall/courses/current/engr207b/continuous_random_variables_2005_04_13_02.pdf

I guess the most abstract setting for a notion of induced probability
would be a measurable function f from a probability space X into a
measure space Y.  The measurable function then induces a probability
distribution on the measure space by:

  Pr_Y(A) = Pr_X(f^-1(A))

where A is any measurable subset of Y, and f^-1(A) is the (measurable)
preimage of A under f.

Hope this helps!

regards, mathtalk-ga

Thanks!