每日彙整: 2017-03-08

樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰生成函數《十後》

當我們說兩個獨立二項分布隨機變數 X, Y 之和是 X + Y 時，意指什麼呢？難到會不是 X 取 x  、 Y 取 y 時， x+y 有意義嗎？就像講如果男人與女人的身高分布都是常態分佈，那麼人的身高分布，當然是講或是男或是女的『混合分布』！它也不必是常態分布的吧！！

所以不管豆子的『顏色』是什麼，

 

每顆豆子都符合豆子機

Bean machine

The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton[1] to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution. Among its applications, it afforded insight into regression to the mean or “regression to mediocrity”.

A working replica of the machine (following a slightly modified design)

 

的機率分布法則︰

Distribution of the balls

If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient  {n \choose k}. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals {n \choose k}p^{k}(1-p)^{n-k}. This is the probability mass function of a binomial distribution.

According to the central limit theorem (more specifically, the de Moivre–Laplace theorem), the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.

 

不同『顏色』只能是『混合』乎？！它的『加法』不知何謂耶！？

因此重要概念之理解務須審慎也☆

Mixture distribution

In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution.

In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions. The individual distributions that are combined to form the mixture distribution are called the mixture components, and the probabilities (or weights) associated with each component are called the mixture weights. The number of components in mixture distribution is often restricted to being finite, although in some cases the components may be countably infinite. More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated under the title of compound distributions.

A distinction needs to be made between a random variable whose distribution function or density is the sum of a set of components (i.e. a mixture distribution) and a random variable whose value is the sum of the values of two or more underlying random variables, in which case the distribution is given by the convolution operator. As an example, the sum of two jointly normally distributed random variables, each with different means, will still have a normal distribution. On the other hand, a mixture density created as a mixture of two normal distributions with different means will have two peaks provided that the two means are far enough apart, showing that this distribution is radically different from a normal distribution.

Mixture distributions arise in many contexts in the literature and arise naturally where a statistical population contains two or more subpopulations. They are also sometimes used as a means of representing non-normal distributions. Data analysis concerning statistical models involving mixture distributions is discussed under the title of mixture models, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions.

Finite and countable mixtures

Given a finite set of probability density functions p1(x), …, pn(x), or corresponding cumulative distribution functions P1(x), …, Pn(x) and weights w1, …, wn such that wi ≥ 0 and ∑wi = 1, the mixture distribution can be represented by writing either the density, f, or the distribution function, F, as a sum (which in both cases is a convex combination):

  F(x)=\sum _{i=1}^{n}\,w_{i}\,P_{i}(x),
  f(x)=\sum _{i=1}^{n}\,w_{i}\,p_{i}(x).

This type of mixture, being a finite sum, is called a finite mixture, and in applications, an unqualified reference to a “mixture density” usually means a finite mixture. The case of a countably infinite set of components is covered formally by allowing  n=\infty \!.

Density of a mixture of three normal distributions (μ = 5, 10, 15, σ = 2) with equal weights. Each component is shown as a weighted density (each integrating to 1/3)

Examples

Simple examples can be given by a mixture of two normal distributions.

Given an equal (50/50) mixture of two normal distributions with the same standard deviation and different means (homoscedastic), the overall distribution will exhibit low kurtosis relative to a single normal distribution – the means of the subpopulations fall on the shoulders of the overall distribution. If sufficiently separated, namely by twice the (common) standard deviation, so  \left|\mu _{1}-\mu _{2}\right|>2\sigma , these form a bimodal distribution, otherwise it simply has a wide peak.[8] The variation of the overall population will also be greater than the variation of the two subpopulations (due to spread from different means), and thus exhibits overdispersion relative to a normal distribution with fixed variation  \sigma , though it will not be overdispersed relative to a normal distribution with variation equal to variation of the overall population.

Alternatively, given two subpopulations with the same mean and different standard deviations, the overall population will exhibit high kurtosis, with a sharper peak and heavier tails (and correspondingly shallower shoulders) than a single distribution.

Univariate mixture distribution, showing bimodal distribution

 

Multivariate mixture distribution, showing four modes

 

事實上那個混色豆子之生成函數仍舊是二項分布的哩☆☆