Derivation of the univariate normal distribution（一元正态分布的推导）.pdfVIP

Derivation of the univariate normal distribution This derivation is a slight adaption from the one given by Dan Teague. Here we elaborate all the steps and incorporate the notion of the expected value. Before we can start the derivation we should first state some mild assumptions. Consider that we are aiming at the origin of the Cartesian plane with darts. We assume the following: 1. The deviation of the darts does not depend on the orientation of the coordinate system. 2. Deviations in orthogonal directions are independent. This means if we deviate a lot in one direction the probabilities of the other direction are not influenced. 3. Large deviations are less likely than small deviations. Consider that we are throwing an dart, the probability that it falls in the interval , under the assumption that is very small (almost infinitesimal), is given by: For the perpendicular direction this is exactly the same, giving . Figure 1 shows this situation, where the shaded area is particular bounded region of which we want to calculate the probability. Under the independence assumption of perpendicular direction we can state that the probability of the bounded region is . Figure 1: Cartesian plane It can of course happen that we are not aiming at the origin but instead aim at an offset of the origin identified by the constant . This way the probability of our translated bound region becomes: We have assumed that the orientation doesn’t matter, and that any similar region r units away from the origin with the area has the same probability. We can state: This implies that Where the function is not dependent on the angle, as every probability is the same. Figure 1 also illustrates a circle with radius r where all the values of the function are the same. Hence this function is only dependent on the radius. For the values and