When we are trying to use Monte Carlo simulation to solve a problem that does not have analytical solution, how do we decide what should be the underlying distribution from which we draw these random numbers?

The examples I find use normal distribution, but when should we use what type of distribution?

  • 5
    $\begingroup$ You should the distribution that describes the underlying process. There's no way to say what it is without knowing what process you're dealing with $\endgroup$
    – Aksakal
    Commented Oct 28, 2015 at 20:46

1 Answer 1


When one uses Monte Carlo methods to approximate an integral$$\mathfrak{I}=\int_\mathcal{X} H(x)\,\text{d}x$$there is a wide range of possible choices for the distribution used in the simulation. The reason is that $\mathfrak{I}$ can be represented in an infinite number of ways as$$\mathfrak{I}=\int_\mathcal{X} h(x)f(x)\,\text{d}x$$where $f$ is a probability density and $h$ is defined as $h(x)=H(x)/f(x)$. The requirements on $f$ are that

  1. the distribution associated with $f$ can be easily simulated, i.e. that there exists an available simulation algorithm (which may rely on MCMC or SMC methods);
  2. the support of $f$ contains the support of $H$;
  3. the resulting Monte Carlo approximation$$\frac{1}{N}\sum_{i=1}^N h(x_i)$$ has a finite variance, which is equivalent to$$\int_\mathcal{X} h^2(x)f(x)\,\text{d}x=\int_\mathcal{X} \frac{H^2(x)}{f(x)}\,\text{d}x<\infty$$

This still leaves open a wide variety of choices, which may be compared through their variance corrected by the computing cost.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.