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{\left(1-\lambda x\right)}
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

有一個數組，相似於：{{'Canada', 3}, {'USA', 5}, {'UK', 2}, {'Brasil', 3}}, 數組的類型是Country, 有兩個變量， Country.name, Country.weight. 每一個國家都有一個權重，而後給一個output()函數，每次調用這個函數的時候就輸出一個國家的名字，要使每一個國家被輸出的機率相等。我用的方法是平攤weight: {Canada, Canada, USA, USA, USA, USA, UK, UK, Brasil, Brasil, Brasil}, 而後用Random 函數輸出。Follow up : 若是這個權重的值很大很大，好比billio級別，應該怎麼辦。個人方法是相似於線段樹，而後再用sum * Random(), 看這個區間坐落在哪裏。

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