Monte Carlo method

Remarks

Example: $q = \pi/4$ is the probability that a point $(U, V)$ uniform on $[0, 1]^2$ satisfies $U^2 + V^2 \le 1$. So $Y = 1$ if $U^2 + V^2 \le 1$ and $Y = 0$ otherwise satisfies clause 1, since $Y$ is bounded and $\mathbb E Y = \pi/4$, and independent uniform points give clause 2. Non-example: $Y = 1/U$ with $U$ uniform on $(0, 1)$ has $\mathbb E\abs{Y} = \int_0^1 u^{-1}\, du = \infty$, so clause 1 fails and the averages $\hat q_n$ tend to $+\infty$ almost surely. The estimate $\hat q_n$ is unbiased, $\mathbb E \hat q_n = q$, and if $\operatorname{Var} Y = \sigma^2 < \infty$ its standard deviation is $\sigma/\sqrt{n}$. In practice clause 2 is realised with a pseudorandom number generator, so the samples are only imitations of independent random variables. Methods based on Markov chains, such as Markov chain Monte Carlo, fit the general sense but not the plain one, since their samples are dependent.