The immediate goal of this chapter is to introduce you to the normal distribution, the central limit theorem, and the t-distribution. These ideas will be developed more later on. By assuming that the sample is at least fairly representative of the population, the sampling distribution can be used as a link between the sample and the population so you can make an inference about some characteristic of the population. The samples will vary, some being highly representative of the population, most being fairly representative, and a few not being very representative at all. Remember that the sample you are using to make an inference about the population is only one of many possible samples from the population. Briefly, the general model of inference-making is to use statisticians’ knowledge of a sampling distribution like the t-distribution as a guide to the probable limits of where the sample lies relative to the population. Since you will be learning to make inferences like a statistician, try to understand the general model of inference making as well as the specific cases presented. The way the t-distribution is used to make inferences about populations from samples is the model for many of the inferences that statisticians make. The t-distribution and the central limit theorem give us knowledge about the relationship between sample means and population means that allows us to make inferences about the population mean. In between discussing the normal and t-distributions, we will discuss the central limit theorem. This makes the t-distribution useful for making many different inferences, so it is one of the most important links between samples and populations used by statisticians. It turns out that t-statistics can be computed a number of different ways on samples drawn in a number of different situations and still have the same relative frequency distribution. The relative frequency distribution of these t-statistics is the t-distribution. For each sample, the same statistic, called the t-statistic, which we will learn more about later, is calculated. The t-distribution can be formed by taking many samples (strictly, all possible samples) of the same size from a normal population. This chapter will discuss the normal distribution and then move on to a common sampling distribution, the t-distribution. If you ever took a class when you were “graded on a bell curve”, the instructor was fitting the class’s grades into a normal distribution-not a bad practice if the class is large and the tests are objective, since human performance in such situations is normally distributed. The normal distribution is the bell-shaped distribution that describes how so many natural, machine-made, or human performance outcomes are distributed. It is normal because many things have this same shape. The normal distribution is simply a distribution with a certain shape. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.Chapter 2.The convolution of a function with a Gaussian is also known as a Weierstrass transform. More generally, if the initial mass-density is φ( x), then the mass-density at later times is obtained by taking the convolution of φ with a Gaussian function. Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time t will be given by a Gaussian function, with the parameter a being linearly related to 1/ √ t and c being linearly related to √ t this time-varying Gaussian is described by the heat kernel. Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), a partial differential equation that describes the time evolution of a mass-density under diffusion.In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. denotes the modified Bessel functions of integer order.
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