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 Subject: Inuced probabilities in statistics Category: Science > Math Asked by: danae05-ga List Price: \$5.00 Posted: 25 Jun 2005 20:56 PDT Expires: 25 Jul 2005 20:56 PDT Question ID: 537046
 ```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```
 danae05-ga rated this answer: `Thanks!`