Given is following function which is also a density
$$f(x)=\left\{\begin{matrix} \frac{3}{4}(2x-x^2) \text{ if } x \in (0,2)\\ 0 \,\,\,\,\,\,\,\text{ else } \end{matrix}\right.$$
Create a random variable with this density.
I ask this question because they can ask similar in exam and I like to know how to do it good and correct?
I think what would be good to use here is rejection sampling. I read about it on Wikipedia but not sure how use it here but nothing else comes to my mind to solve it.
Formula I find is $$M := \sup_{x}\frac{f(x)}{g(x)} < \infty$$
I sum up algorithm because on Wikipedia it's very long:
- Simulate $U \sim R(0,1)$, in case you don't know this notation: It means that random variable $U$ is equally / uniformly distributed from interval $0$ to $1$
- Simulate $Y \sim g$
- Accept $X=Y$ if $U \leq \frac{1}{M} \frac{f(Y)}{g(Y)}$, else go to step 1. (repeat it)
- We have $P(X<x) = \int_{0}^{x} dt = x$ so $X$ has density function
$$f(x)=\left\{\begin{matrix} 1 \text{ if }0 \leq x < 1\\ 0 \text { else } \end{matrix}\right.$$
I hope this is correct for first step? But I don't know how to do second step and that's why third I can't do too : /
Maybe there is other method for solving this? Because I only know about this.
I'd advise you to read some practical examples on how to do rejection sampling.
Just wanted to say, the CDF ($CDF(x)=\int_0^xf(x')dx'$) of your probability function can actually be inverted. And if you fill in your $x\sim~Uniform[0,1]$ into the inverted CDF, you get a random variable distributed like your $f(x)$. The method is explained here: https://en.wikipedia.org/wiki/Inverse_transform_sampling