Induced Measure: Understanding the Concept

[peter.a]

What is an induced measure?
I have seen the formal definition many times i am trying to get a grasp of this concept.

Does an induced measure mean that we can view the measure associated with a random variable as some co-ordinate function defined on R?

Is it the cdf?

 

[Vargo]

Since you have seen the formal definition, let me try to illustrate with a few examples.

You toss a coin three times. The sample space is composed of things that look like HTH. There are 8 elements, and if the coin is fair they all have equal probability. Let X be the random variable that sums the numbers of heads. Now the possible values of X are 0,1,2,3. The induced measure is defined on the values of X and gives values of 1/8, 3/8,3/8,1/8 respectively.

Ok, now suppose the experiment is to throw darts at a dartboard, and let's assume for simplicity that they always hit the board and stick. Then there is some probability distribution defined on the disk that tells the likelihoods of hitting the various points. Now let X be the distance from the bull's eye. The induced measure here is a probability measure on [0,R], where R is the radius of the board.

If you have an experiment with sample space S, and then you have a random variable X, the induced measure is the probability distribution you get when you think of the values of X as the new sample space. Of course, there is a much more precise definition, but you have read that so I won't repeat it.