Intuitive examples of importance sampling

Question

My background is computer science. I am fairly new to monte carlo sampling methods and, although I understand the math, I have hard time coming up with intuitive examples for importance sampling. More precisely, could someone provide examples of:

1. an original distribution one cannot sample from but one can estimate
2. an importance distribution which can be sampled from and adequate for this original distribution.

Answer 1

Suppose you want to simulate the mean of a standard normal distribution that is truncated to the unit interval $[0,1]$.

An inefficient way would be to take draws from $N(0,1)$, but only keep draws in [0,1]. Then you calculate the mean using only the data you kept.

A more efficient way would be to draw from $U(0,1)$, and calculate the importance weights, which you can use in calculating a weighted mean.