확률분포함수 구현하기

reference)) Roy D. Yartes, and David J. Goodman, Probability and Stochastic Processes, 2nd edition, 2005, pp.135~144.

Random Samples

1) 평균과 분산을 이용하여 구하는 방법
Conveniently, MATLAB also includes the built-in function randn to generate random samples of standard normal random variables. Thus We generate Gaussian (μ,σ) random variables by stretching and shifting standard normal random variables 

function x = gaussrv(mu, sigma, m)  x = mu + (sigma*randn(m,1));

 

2) CDF의 역행렬을 이용하여 구하는 방법
For continuous random variables, we use following theorem to transform a uniform (0,1) random variable U into other types of random variables.

예1) X가 exponential random variable인 경우 (즉, F_X(x)=1-e^-x   x≥0)에 대한 sample를 발생

=> Note that if u=F_X(x)=1-e^-x, then x=-ln(1-u).
if U is a uniform (0,1) random variable, then Y=-ln(1-U) is an exponential (λ=1) random variable.
Using rand to approximate U, we have the following MATLAB code:

function x = exponentialrv(lambda, m)  x = -(1/lambda) * log(1-rand(m,1);

 

예2) θ가 [0,pi] 범위에서 pdf가 W(θ)=0.5*sinθ인 경우

=> ori_the = acos(1-2*rand(1,N));      

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The following theorem shows how to derive various types of random variable from the transformation X=g(U) where U is a uniform (0,1) random variable.

Theorem]
Let U be a uniform (0,1) random variable.
Let F(x) denote a CDF (cumulative distribution function) with an inverse F^-1(u) defined for 0<u<1.
=> The random variable X=F^-1(u) has CDF F_X(x)=F(x).

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간단히 말해서 X의 CDF (F_X(x))와 F(x)의 역행렬의 x축 값이 uniform random variable인 경우의 CDF는 같다.

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The technique of theorem is particularly useful when the CDF is an easily invertible function.
Unfortunately, there are many cases, including Gaussian and Erlang random variables, when the CDF is difficult to compute much less to invert.

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We use this technique with the MATLAB rand function approximating U to generate sample values of a random variable X.