Pseudo Random Nubmer Sampling

Pseudo Random Nubmer Sampling

https://en.wikipedia.org/wiki/Inverse_transform_sampling

given a distribution's cumulative distribution function (CDF), generate sample numbers for this distribution.

typically based on uniform distribution variable X (or several of them), then somehow manipulate it, and get random variable Y which has the required distribution

Rejection Sampling if density function is known

one type of Monte-Carlo Method
see some notes

target: sample from F=f(x)

idea: find an alternative G=g(x) which we already know, and that f(x)/g(x) <= c where c is a constant (ideally close to 1)

algorithm:

1. sample y from G;
2. sample u from U[0,1];
3. if u <= f(y)/c*g(y), then accept y; reject otherwise

Inverse Transform Sampling for distributions where CDF is known

1. input 1: CDF of some distribution; for example, exponential distribution, F(x)=1-exp
2. input 2: a uniform distribution U[0,1]; for example, u=0.387;
3. F(x) = y => x = F^{-1}\left(y\right) = -\frac{1}{\lambda}\ln{\left(1-y\right)} => x = -\frac{1}{\lambda}\ln\left(y\right)
4. draw a value from U[0,1], and use it as CDF() value, then solve for the corresponding x value

Box-Muller Transform for Normal Distribution

1. only used for generating Normal Distribution
2. input: uniform distribution U[0,1]
3. output: 2 independent standard normal distribution numbers
4. Suppose U1 and U2 are independent random variables from U[0,1]
5. let and , then Z0 and Z1 are both N(0,1) random variables

example

有一个数组，类似于：{{'Canada', 3}, {'USA', 5}, {'UK', 2}, {'Brasil', 3}}, 数组的类型是Country, 有两个变量， Country.name, Country.weight. 每个国家都有一个权重，然后给一个output()函数，每次调用这个函数的时候就输出一个国家的名字，要使每个国家被输出的概率相等。我用的方法是平摊weight: {Canada, Canada, USA, USA, USA, USA, UK, UK, Brasil, Brasil, Brasil}, 然后用Random 函数输出。Follow up : 如果这个权重的值很大很大，比如billio级别，应该怎么办。我的方法是类似于线段树，然后再用sum * Random(), 看这个区间坐落在哪里。

1. target distribution is a discrete distribution, p(x='Canada')=3/13, p(x='USA')=5/13 etc.
2. fit it into the Inverse Transform Sampling algorithm
3. sample an integer from [1,13], {1,2,3} => Canada, {4,5,6,7,8} => USA, {9,10} => UK, {11,12,13} => Brasil