Friday, August 15, 2008
I was looking at a CD holder and noticed that it could represent the unit circle and provide a model of a characteristic function. It reminded me of a paper by Epps, available here, whose abstract begins: "The value of a characteristic function of a random variable X at some real number t is the center of mass of the distribution of tX wrapped around the unit circle in the complex plane." I have cut the sides of the CD holder to leave a normal distribution wrapped around a circle. The balance point of this model is the center of mass mentioned by Epps. As t gets smaller, approaching zero, the wrapped distribution gets more concentrated around a modal point. Also, the balance point moves toward the edge of the unit circle, directly under the mode of the wrapped distribution. The rate at which this balance point moves, at the edge of the unit circle, is the mean of the the random variable X.