I am looking for some non-uniform probability distributions that cannot be solved analytically (if there are). I am not Math or Statistics major so I hope someone can guide me to some help sources.
I am aware that, in simulation, they transform uniformly distributed random numbers to the non-uniform random numbers distribution that they desire. If one find it hard/time consuming to solve it analytically, they use sampling method to convert the uniform random numbers to the specific non-uniform distributed random numbers they want. I wonder, what are those distributions that cannot be solved analytically? Or what are the disadvantages using sampling method?
For clarification, the reason I ask this is, we have the ability to generate random numbers (hardware) with arbitrary distributions, I just want to learn about potential applications of our system. Thank you
To generate a non-uniform distribution you can use following 3 methods. All these methods aim to generate a probability distribution from a uniformly distributed random variable $u$.
Inverse Transformation Method
This is the method which you are most likely referring to when you say "solve analytically".
Say you want to generate the cumulative distribution $F(x)$ using $u$. Let's assume $F(x)$ is continuous. $F(x)$ is monotonically increasing and bounded. So we can find an inverse of $F(x)$. The inverse is given by $$x=F^{-1}(u)\quad\text{such that}\quad 0\le u\le1$$ This generates the random variable $X$ with cdf $F(x)$.
However this method has an issue that it might be difficult to find an inverse analytically. You could also solve the equation $F(x)-u=0$ but it is time consuming to solve this.
Another good enough solution can be found by taking a table of values of $F(x)$ at various values of $x$, like $F_1, F_2, F_3,..., F_n$. And perform a lookup of $F_{k-1}\le u\le F_k$
Acceptance/Rejection Method
This method uses the distribution function(pdf) instead of cdf. The pdf, $f(x)=F'(x)$. In this method we try to find another pdf $g(x)$ such that $$f(x)\le cg(x)\quad\forall\quad x\in\mathbb{R}$$Now we generate a random number $u$ and accept it if $u\le f(x)/cg(x)$.
In this case the difficulty depends on complexity of $f(x)$ and, $g(x)$, and also on $c$.
Mixture of distribution
In this case we decompose our cdf into multiple cdf for which it is easy to generate a random number. $$F(x)=p_1F_1(x)+p_2F_2(x)...,\quad \sum p_i=1$$ Now we select a random number $i$ with probability $p_i$ and generate a random variable with distribution $F_i(x)$.
This is useful when you can't find a $g(x)$ in using the above method for the entire range.
Mentioned above are the few basic techniques. You can look here for more methods for the same and here for more details on the methods I have mentioned.